
549 F.3d 119 (2008)
UNITED STATES of America, Appellee,
v.
Jaime CHAVEZ, Anastacio Acosta, Defendants-Appellants.
Docket Nos. 05-4679cr, 05-5401cr.
United States Court of Appeals, Second Circuit.
Argued: May 28, 2008.
Decided: December 8, 2008.
*123 Eric Snyder, Assistant United States Attorney, New York, N.Y. (Michael J. Garcia, United States Attorney for the Southern District of New York, Anirudh Bansal, Jonathan S. Kolodner, Assistant United States Attorneys, New York, NY, on the brief), for Appellee.
Robert J. Boyle, New York, N.Y. (Robert Ramsey, Jr., Ramsey & Price, Los Angeles, CA, on the brief), for Defendant-Appellant Chavez.
Joseph W. Martini, Southport, CT (Michael R. Patrick, Pepe & Hazard, Southport, CT, on the brief), for Defendant-Appellant Acosta.
Before: KEARSE, CALABRESI, and SACK, Circuit Judges.
KEARSE, Circuit Judge:
Defendants Jaime Chavez and Anastacio Acosta appeal from judgments entered in the United States District Court for the Southern District of New York following a jury trial before Gerard E. Lynch, Judge, convicting each defendant of conspiracy to distribute and possess with intent to distribute more than five kilograms of cocaine, in violation of 21 U.S.C. § 846 (count one), and convicting Chavez of possession, in furtherance of the cocaine trafficking conspiracy, of a firearm equipped with a silencer, in violation of 18 U.S.C. §§ 924(c)(1)(A) and (c)(1)(B)(ii) (count two). Chavez was sentenced principally to 300 months' imprisonment on count one and to 360 months' imprisonment on count two  the statutory minimum for that count  to be served consecutively to the term imposed for count one, for a total prison term of 660 months. Acosta was sentenced principally to 198 months' imprisonment. On appeal, Chavez and Acosta contend that the evidence was insufficient to support their convictions and that there were trial errors, and they challenge their sentences. Finding no basis for reversal, we affirm. We write principally to address Chavez's contention that the district court erred in concluding that, having determined an appropriate sentence for him on count one, it had no authority to reduce that sentence on account of the severity of the sentence it was required to impose for count two.

I. BACKGROUND
The indictment on which Chavez and Acosta were tried alleged that they were members of a narcotics conspiracy that operated in various areas, including New York City and California, from approximately October 2001 through May 2003. Other alleged members of the conspiracy included Gregorio Barraza, his brothers *124 Daniel Barraza and Jose Luis Barraza, and cooperating witness Nicholas Ibarra. The government's evidence at trial, discussed in greater detail in Part II.A. below, consisted principally of (a) recordings of telephone conversations between Chavez and other conconspirators, and (b) the testimony of Ibarra who, inter alia, described the organization's narcotics distribution operation in New York and interpreted some of the coded terms used in conconspirators' telephone conversations. The government also introduced in evidence a pistol, equipped with a silencer, that had been seized from Chavez's apartment.
The jury found both Chavez and Acosta guilty of conspiracy to distribute and possess with intent to distribute more than five kilograms of cocaine. It found Chavez guilty of possessing the silencer-equipped pistol in furtherance of that drug trafficking conspiracy.
The district court sentenced Acosta principally to 198 months' imprisonment (see Part II.C.2. below). The court sentenced Chavez principally to 300 months' imprisonment on count one, followed by 360 months' imprisonment  the statutory mandatory minimum sentence for his conviction on count two  for a total of 660 months' imprisonment (see Part II.B. below).

II. DISCUSSION
On appeal, both defendants contend principally (1) that the evidence was insufficient to support their convictions, and (2) that their sentences were unreasonable. They also advance various other contentions, including that statements made by the government in summation were improper and that the district court should have given an accomplice-witness instruction in the language requested by Acosta. We find no merit in any of defendants' contentions; only the evidentiary and sentencing challenges warrant discussion.

A. Sufficiency of the Evidence

Both Chavez and Acosta contend that the government's evidence at trial was insufficient to permit the jury to find that they were members of the conspiracy alleged in the indictment. Acosta contends that the evidence showed that there existed not the single California-New York conspiracy alleged, but rather multiple conspiracies, and that he was a member only of the smaller and independent conspiracy that operated in New York. Chavez contends that the evidence failed to show that he had any connection with the conspiracy that operated in New York; he also challenges the sufficiency of the evidence to show that the gun seized from his apartment (a) was possessed by him, and (b) was possessed in furtherance of the drug-trafficking conspiracy.
In challenging the sufficiency of the evidence to support a conviction, a defendant bears a heavy burden. See, e.g., United States v. Quattrone, 441 F.3d 153, 169 (2d Cir.2006); United States v. Matthews, 20 F.3d 538, 548 (2d Cir.1994). In considering such a challenge, we must view the evidence in the light most favorable to the government, crediting every inference that could have been drawn in the government's favor, see, e.g., United States v. Locascio, 6 F.3d 924, 944 (2d Cir.1993), cert. denied, 511 U.S. 1070, 114 S.Ct. 1645, 128 L.Ed.2d 365 (1994), and "defer[ring] to the jury's assessment of witness credibility," United States v. Bala, 236 F.3d 87, 93 (2d Cir.2000), and its assessment of the weight of the evidence, see, e.g., United States v. Morrison, 153 F.3d 34, 49 (2d Cir.1998). The conviction must be upheld if "any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt." Jackson v. *125 Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979) (emphasis in original); see, e.g., United States v. Velasquez, 271 F.3d 364, 370 (2d Cir.2001) (a conviction may be overturned on the basis of insufficiency of the evidence only if, on the evidence viewed in the light most favorable to the government, with all inferences drawn and credibility assessments made in its favor, "`no rational trier of fact could have found the essential elements of the crime charged beyond a reasonable doubt'" (quoting United States v. McDermott, 245 F.3d 133, 137 (2d Cir.2001))). These principles apply whether the evidence being reviewed is direct or circumstantial. See, e.g., Glasser v. United States, 315 U.S. 60, 80, 62 S.Ct. 457, 86 L.Ed. 680 (1942).
In order to convict a defendant of conspiracy, the government must prove both the existence of the conspiracy alleged and the defendant's membership in it beyond a reasonable doubt. See, e.g., United States v. Huezo, 546 F.3d 174, 180 (2d Cir.2008). The essence of any conspiracy is, of course, agreement, and in order to establish a conspiracy, the government must show that two or more persons entered into a joint enterprise with consciousness of its general nature and extent. See, e.g., United States v. Alessi, 638 F.2d 466, 473 (2d Cir. 1980). To establish a particular defendant's membership in the alleged conspiracy, the government must present "proof of [his] purposeful behavior aimed at furthering the goals of the conspiracy." United States v. Diaz, 176 F.3d 52, 97 (2d Cir.) (internal quotation marks omitted), cert. denied, 528 U.S. 875, 120 S.Ct. 181, 145 L.Ed.2d 153 (1999). "Both the existence of a conspiracy and a given defendant's participation in it with the requisite knowledge and criminal intent may be established through circumstantial evidence." United States v. Stewart, 485 F.3d 666, 671 (2d Cir.2007). Further,
[t]he government need not show that the defendant knew all of the details of the conspiracy, "so long as he knew its general nature and extent." United States v. Rosa, 17 F.3d 1531, 1543 (2d Cir. 1994). Nor [need] the government prove that the defendant knew the identities of all of the other conspirators. United States v. Downing, 297 F.3d 52, 57 (2d Cir.2002).
United States v. Huezo, 546 F.3d at 180; see, e.g., Blumenthal v. United States, 332 U.S. 539, 557, 68 S.Ct. 248, 92 L.Ed. 154 (1947) ("[T]he law rightly gives room for allowing the conviction of those discovered upon showing sufficiently the essential nature of the plan and their connections with it, without requiring evidence of knowledge of all its details or of the participation of others.").
The matter of whether there existed a single conspiracy as charged in the indictment or multiple conspiracies "is a question of fact for a properly instructed jury." United States v. Berger, 224 F.3d 107, 114 (2d Cir.2000) (internal quotation marks omitted).
In order to prove a single conspiracy, the government must show that each alleged member agreed to participate in what he knew to be a collective venture directed toward a common goal. The coconspirators need not have agreed on the details of the conspiracy, so long as they agreed on the essential nature of the plan.
United States v. Geibel, 369 F.3d 682, 689 (2d Cir.) (internal quotation marks omitted), cert. denied, 543 U.S. 999, 125 S.Ct. 619, 160 L.Ed.2d 457 (2004). "[A] single conspiracy is not transformed into multiple conspiracies merely by virtue of the fact that it may involve two or more phases or spheres of operation, so long as there is *126 sufficient proof of mutual dependence and assistance." Id. (internal quotation marks omitted). Thus,
[i]n the context of narcotics operations,. . . we have held that even where there are multiple groups within an alleged conspiracy, a single conspiracy exists where the groups share a common goal and depend upon and assist each other, and we can reasonably infer that each actor was aware of his part in a larger organization where others performed similar roles.
United States v. Berger, 224 F.3d at 115 (internal quotation marks omitted); see also United States v. Sureff, 15 F.3d 225, 230 (2d Cir.1994) ("A single conspiracy may encompass members who neither know one another's identities, . . . nor specifically know of one another's involvement. . . .").
Here, the district court properly instructed the jury on single versus multiple conspiracies  defendants do not contend otherwise  and we conclude that there was sufficient evidence for the jury to conclude that the single conspiracy alleged in the indictment, operating in California and New York, existed and that both Chavez and Acosta were members of it. Taken in the light most favorable to the government, the evidence included proof of the following.

1. Defendants' Membership in the Conspiracy

Chavez, who operated primarily out of California, supervised the organization's operations in the United States; Chavez reported to his uncle Santiago Chavez-Ayon, known as "Santi," who was based in Mexico. The organization's operations in New York were overseen by Gregorio Barraza ("Barraza"). Some of the narcotics sold by the organization came from Mexico; some came from suppliers in the eastern United States, including Acosta.
Ibarra testified that, prior to moving to New York, he had been friendly with the Barraza brothers when he and they lived in Compton, California (see Trial Transcript ("Tr.") at 382), and he had met Chavez casually in California (see id. at 417-18). Ibarra began working with Barraza in New York City in the spring of 2002, engaging in drug trafficking with Barraza, Acosta, and "a lot of [other] people" (id. at 368). Barraza was in charge of the New York group; Ibarra understood that Barraza's boss was Chavez. (Id. at 368-69.) Indeed, at one point, when Barraza discovered that several kilograms of cocaine were missing and suspected that Ibarra, high on cocaine and marijuana, had misdelivered them, Barraza said that if Chavez saw Ibarra "all drugged up" Chavez would fire Ibarra. (Id. at 424-25.)
Under the direction of Barraza, Ibarra began living in a stash house maintained by the organization in Queens, New York. Ibarra's jobs included picking up cocaine from suppliers, delivering it to the organization's distributors, and keeping count of the drug money stored in the stash house.
Acosta supplied the organization with wholesale quantities of cocaine. For example, one of Ibarra's early tasks in New York was to drive in tandem with Barraza to Acosta's house in the Bronx, New York, where Acosta and Barraza gave Ibarra a duffel bag containing 50 kilograms of cocaine. Ibarra temporarily stored the cocaine at the stash house, and over the next few days he delivered bulk quantities to the organization's distributors. Some days thereafter, Ibarra collected sales proceeds from the distributors and delivered money to Acosta. The proceeds of these sales amounted to "more than several hundred thousand dollars." (Tr. 401-02.)
*127 This process was repeated several times, with various quantities of cocaine (see generally Part II.C.1. below) supplied by Acosta. In addition, on at least one occasion, Ibarra picked up 40 kilograms of cocaine from a supplier other than Acosta, using a vehicle provided by Acosta.
While working with Barraza in New York, Ibarra fielded numerous telephone calls for Barraza from Chavez in California. The government, which conducted court-authorized wiretaps on Chavez's telephone in California, Barraza's telephone in New York, and Ibarra's cell phone, introduced many recordings and transcripts of telephone conversations between Chavez and Barraza in which Chavez gave instructions for or expressed concern over, inter alia, the New York operation's inventory and security.
For example, in early August 2002, concerned that the organization's assets had been depleted, Chavez called Barraza and asked whether Barraza could "put anything together right now" (August 8, 2002 call, GX 15T, at 2). Barraza replied that he had seven kilograms of cocaine; Chavez instructed him to dilute it so that "out of those seven" they could get "fourteen." (Id.)
In mid-August, Chavez called Barraza and informed him that the police had paid a visit to Barraza's mother's house in Compton, "looking for all of us" (August 19, 2002 call, GX 18T, at 3). Chavez said the police were looking for Barraza, his two brothers, and several others, including Acosta. (See id. at 2.) Chavez instructed Barraza to "get out" and hide at a ranch in Mexico "for a while." (Id. at 5.)
Two days later, Chavez reported to his uncle Santi that "we're all in hiding" because law enforcement agents had a list of people for whom they were searching, including Daniel Barraza and Chavez. (August 21, 2002 call, GX 21T, at 4.) Santi asked whether the list also included Acosta. Chavez said it did but that the agents were "looking for him here" in California, as they apparently did not realize that Acosta was in New York. (Id. at 4; see also Tr. 377 (in the late 1990s, Acosta distributed cocaine in Compton, California).)
In mid-August, the organization had moved its New York inventory from Queens to a new stash house in the Bronx. It then decided that it needed to change locations again; and in a conversation with Barraza, Chavez expressed concern about when the new stash house would be ready (August 29, 2002 call, GX 24T, at 1-2).
In the fall of 2002, law enforcement agents in New York made several seizures of cocaine from the organization. In late October they arrested Jose Nunez (also known as "Che") (see, e.g., Tr. 783-85), one of the organization's New York distributors to whom Ibarra regularly made bulk deliveries of cocaine supplied by Acosta (see, e.g., id. at 375, 400, 409-12). Che had an auto body shop in the Bronx (see id. at 412), and in connection with his arrest at that location a total of 24 kilograms of cocaine and $120,000 in cash were found concealed in vehicles (see id. at 787). An expert witness for the government testified that when a narcotics distribution organization is informed of such an arrest and seizure, the organization's leaders typically demand to see documentary proof of the seizure  commonly "legitimate press clippings or legitimate court documents"  in order to be assured that the drugs have not in fact been stolen by the arrested member. (Id. at 759-61.) Following Che's arrest, Chavez and Barraza discussed the need to get Che's "papers" (November 14, 2002 call, GX 56T, at 2-3).
A month before Che's arrest, agents had arrested another of the organization's distributors *128 and seized a duffel bag that had just been delivered to him by Ibarra. The bag contained approximately five kilograms of cocaine. Thereafter, the agents arrested Ibarra at the organization's stash house, from which they seized, inter alia, 25 kilograms of cocaine.
Ibarra testified that in the fall of 2002, the price of a kilogram of cocaine was approximately $23,000. The cocaine seizures described above, totaling 54 kilograms, were thus worth more than $1.2 million.
In a call to Barraza in the spring of 2003, Chavez stated that, in the seizures from the New York operation, Chavez had "lost everything" (May 1, 2003 call, GX 62T, at 4 ("I lost . . . everything that I could possibly lose. . . .")). Chavez said, "I'm desperate" and going "crazy . . . from all the effort that I'm making to cover that debt." (Id.)
In the same conversation, Chavez said he had heard a rumor that Ibarra, arrested in September 2002, had been released from jail. Chavez expressed concern that Ibarra's release would make the organization's suppliers "think that we lied" about the cocaine's having been seized (id. at 3) and lead them to take violent action against Chavez (see id. at 4 ("[T]he same thing that happened to Abraham is going to happen to us. You know that fool Abraham? The one who got killed?")).
Thus, the recorded telephone conversations between Chavez and Barraza, along with the testimony of Ibarra, showed, inter alia, that Barraza was in charge of the organization's operations in New York; that Acosta supplied the organization's New York operation with cocaine; that Chavez supervised Barraza with respect to the New York operation's inventory and its security, including the storage of cocaine, the detection of theft by employees, and the avoidance of apprehension by law enforcement agents; that in identifying the New York personnel who should be aware of government attention, Chavez expressly included Acosta; and that Chavez knew that he himself was personally responsible for the "debt" to the organization's suppliers resulting from the cocaine seizures in New York. This provided ample evidence from which the jury could infer that Chavez, from California, and Acosta in New York, were members of the conspiracy whose operations included the Barraza-supervised distribution in New York of cocaine supplied by Acosta.
In addition, the record included evidence of Chavez's involvement in the New York distribution operation prior to Ibarra's 2002 arrival in New York and the interception of Chavez's telephone conversations. In October 2001, Chavez and Jose Luis Barraza attempted to drive from New York to California with a total of $100,000 in U.S. currency in bags or concealed in shampoo and conditioner bottles inside a hidden compartment in their car. (See Tr. 816-17, 827-39.) The car was stopped by police in Omaha, Nebraska, for traffic violations, and the police found the money during consensual searches of the car. (See id. at 806-11, 827-28.) The narcotics-detection dog used by the Omaha police indicated that the hidden compartment smelled of narcotics. (See id. at 835-36.)
We note that Chavez complains of the admission of the report of this seizure on the ground that the government did not give timely notice of its intention to introduce the seizure report in its case-in-chief, see Fed.R.Crim.P. 16(a)(1)(E)(ii). Although the government did not give such notice until a week before the trial was scheduled to begin, the court found that the government's tardiness was inadvertent and that Chavez was not entirely without warning, given that *129 the Omaha seizure was described in an affidavit that was produced to Chavez in discovery some 15 months before trial. The court concluded that Chavez would not be prejudiced by the delay because the court would afford him, without the usual formalities, a hearing on a motion to suppress the seizure report. Following that hearing, the court determined that the search was proper and that the evidence was admissible. Chavez does not contend that any of the district court's factual findings or conclusions of law were erroneous; he argues only that the seizure report should have been automatically excluded on the ground that the government's notice of intent to introduce the report was late. We disagree. The trial court has broad discretion to fashion a remedy for the government's violation of Rule 16(a), see generally United States v. Thai, 29 F.3d 785, 804 (2d Cir.), cert. denied, 513 U.S. 977, 115 S.Ct. 456, 130 L.Ed.2d 364 (1994), and we conclude that, in all the circumstances, the district court's decision to admit the report following the hearing was well within the proper bounds of its discretion. Accordingly, the evidence that Chavez was engaged in transporting to California what were likely the proceeds of narcotics transactions in New York was properly admitted and was further evidence from which the jury could infer that the wide-spread conspiracy alleged in the indictment existed and that Chavez was a member of it.

2. Chavez's Gun Possession in Furtherance of the Conspiracy

Chavez also contends that the evidence was insufficient to support his conviction on count two, i.e., possession of a firearm found in his apartment, equipped with a silencer, in furtherance of the count-one narcotics conspiracy of which he was convicted, see 18 U.S.C. § 924(c)(1)(A) (prohibiting possession of a firearm for such a purpose), and id. § 924(c)(1)(B)(ii) (prescribing enhanced punishment if the firearm is equipped with a silencer). Chavez challenges the sufficiency of the evidence as to the elements of possession and furtherance.
In order to establish that a defendant possessed a firearm within the meaning of § 924(c), the government need not prove that he physically possessed it; proof of constructive possession is sufficient. See, e.g., United States v. Finley, 245 F.3d 199, 203 (2d Cir.2001), cert. denied, 534 U.S. 1144, 122 S.Ct. 1101, 151 L.Ed.2d 997 (2002); cf. United States v. Rivera, 844 F.2d 916, 925-26 (2d Cir.1988) (applying constructive possession principles in upholding conviction under the predecessor to 18 U.S.C. § 922(g), which, inter alia, made it unlawful for a convicted felon to possess a firearm). Constructive "[p]ossession of a firearm may be established by showing that the defendant knowingly [had] the power and the intention at a given time to exercise dominion and control over [it]." United States v. Finley, 245 F.3d at 203 (internal quotation marks omitted); see, e.g., United States v. Rivera, 844 F.2d at 925; United States v. Tribunella, 749 F.2d 104, 111-12 (2d Cir. 1984) (where there was evidence that the defendant's other family members knew nothing about guns and that the defendant owned guns and had ammunition and a magazine about guns in his bedroom, the evidence was sufficient to support a finding that he constructively possessed a gun found above the ceiling in his bedroom).
In the present case, the record included the following evidence. During the spring of 2003, Chavez was living in an apartment on Petrol Street in Paramount, California. On May 9, law enforcement agents observed him entering the apartment wearing a gray T-shirt with a black design. *130 When he exited later that day, he went to a nearby motel where he spent the night. He was arrested the next morning, and he gave his consent for a search of the apartment.
Although Chavez argues, inter alia, that no inferences could be drawn as to his possession of items found in the two-bedroom, two-bathroom apartment because his roommate had  and could give others  access to the apartment, Chavez had recently told Barraza that he was living in the Petrol Street apartment with a roommate who was "never here." (April 29, 2003 call, GX 60T, at 7.) Consistent with Chavez's statement to Barraza, the agents who searched the apartment on the day of the arrest testified that only one of the bedrooms and one of the bathrooms showed any signs of occupancy. (See, e.g., Tr. 924 (in one bedroom "[t]here weren't any linens on the bed, no clothing in the closet"; in one bathroom "[t]here weren't any toiletries" or "any towels or anything like that").)
The other bedroom appeared to be lived-in. There was a large bed with linens on it; there were clothes in the closet and shoes on the floor; and in the adjacent bathroom were such items as toiletries and towels. (See id. at 924-25.) In that bedroom, the agents found a .22-caliber Beretta pistol under a pillow on the bed. (See id. at 926.) On top of that bed was the gray and black T-shirt that Chavez had worn the day before. (See id. at 894-95, 907-09.)
This evidence was ample to permit the jury to find that Chavez had dominion and control over the pistol found under the pillow on his bed and hence that he possessed it within the meaning of 18 U.S.C. § 924(c)(1)(A).
A gun may, of course, be possessed for any of a number of purposes, some lawful, others unlawful. The government does not establish that a firearm was possessed in furtherance of drug trafficking merely by relying on the proposition that drug dealers generally use guns to protect themselves and their drugs, and thus that any time a gun is possessed by a drug dealer it is possessed in furtherance of his drug offenses. See, e.g., United States v. Snow, 462 F.3d 55, 62 (2d Cir. 2006), cert. denied, 549 U.S. 1150, 127 S.Ct. 1022, 166 L.Ed.2d 770 (2007). "Instead, the government must establish the existence of a specific `nexus' between the charged firearm and the [federal drug trafficking crime]." Id.; see, e.g., United States v. Lewter, 402 F.3d 319, 321-22 (2d Cir.2005); United States v. Finley, 245 F.3d at 203. The "nexus" inquiry is fact-intensive.
"Although courts look at a number of factors to determine whether such a nexus exists," United States v. Snow, 462 F.3d at 62, such as "the type of drug activity that is being conducted, accessibility of the firearm, the type of the weapon, whether the weapon is stolen, the status of the possession (legitimate or illegal), whether the gun is loaded, proximity to drugs or drug profits, and the time and circumstances under which the gun is found," id. at 62 n. 6 (internal quotation marks omitted), "the ultimate question is whether the firearm `afforded some advantage (actual or potential, real or contingent) relevant to the vicissitudes of drug trafficking,'" id. at 62 (quoting United States v. Lewter, 402 F.3d at 322). "Thus, while no conviction would lie for a drug dealer's innocent possession of a firearm,. . . a drug dealer may be punished under § 924(c)(1)(A) where the charged weapon is readily accessible to protect drugs, drug proceeds, or the drug dealer himself." United States v. Snow, 462 F.3d at 62-63 (internal quotation marks omitted).
*131 Notwithstanding Chavez's argument that the in-furtherance element was not proven here because he was not carrying the gun and the agents found no narcotics, money, or narcotics paraphernalia in the apartment, the record sufficed to permit the jury to find that there was a nexus between the narcotics conspiracy of which Chavez was a member and Chavez's possession of the gun. It included the following.
The Beretta pistol found under Chavez's pillow was loaded with hollow-point bullets, which are designed to expand upon impact and hence to create injury beyond the effects caused by mere impact. (See Tr. 964-65.) The pistol was equipped with a silencer that bore no manufacturer's name or serial number, thereby indicating that it had been fabricated clandestinely. (See id. at 973-74.) Thus, the nature of both the ammunition and the silencer were indicia that Chavez's pistol was possessed for unlawful purposes. See, e.g., United States v. Lewter, 402 F.3d at 322 (gun's obliterated serial number and hollow-point bullets were indicia of possession for an unlawful purpose). And there was postarrest evidence indicating that Chavez in fact possessed the pistol for use in connection with the narcotics conspiracy: While Chavez and coconspirators Barraza, Daniel Barraza, and Jose Luis Barraza were in adjacent holding cells following their arrests, one of the arresting agents heard one of the Barraza brothers yell out a question to Chavez, "did they found [sic] the gun[?]" (Tr. 116).
Finally, the gun was seized just days after Chavez told Barraza, as discussed in Part II.A.1. above, that Chavez feared that their suppliers, believing Chavez had stolen the cocaine and would not pay the debt, would attempt to kill him as they had killed "that fool Abraham." (May 1, 2003 call, GX 62T, at 4.)
In sum, the evidence was ample to permit the jury to find that Chavez possessed the silencer-equipped gun found under his pillow to provide security for his narcotics conspiracy operation and protect Chavez against an expressly anticipated murder attempt on his life by his drug suppliers, and hence was possessed in furtherance of his drug trafficking crime.

B. Chavez's Contention that the District Court Misapprehended Its Sentencing Authority

As Chavez was convicted on count one of conspiring to traffic in more than five kilograms of cocaine, and had been convicted of a drug-trafficking felony previously, the district court was required to sentence him to, inter alia, a prison term of at least 20 years (240 months). See 21 U.S.C. §§ 841(a), 841(b)(1)(A), 846. The district court, consulting the 2004 version of the Sentencing Guidelines, determined that Chavez's total offense level was 42, comprising a base offense level of 38 because Chavez was responsible for conspiring to distribute more than 150 kilograms of cocaine and a four-step increase because he was a leader of criminal activity involving five or more participants. As Chavez's record of convictions placed him in criminal history category III, the advisory-Guidelines-recommended range of imprisonment for him on count one was 360 months to life.
With respect to count two, on which the jury found that the Beretta pistol discovered under Chavez's pillow was possessed by Chavez in furtherance of his drug-trafficking conspiracy and was equipped with a silencer, the court was required to sentence Chavez to a prison term of not less than 30 years (360 months) in addition to the prison term imposed on count one. See 18 U.S.C. §§ 924(c)(1)(A), (c)(1)(B)(ii), and (c)(1)(D)(ii). Chavez's attorney acknowledged *132 that the court was required by the pertinent statutes to impose at least a 240-month prison term on count one plus 30 years on count two, and that thus "we're looking at minimum[, at a] mandatory 50 years" (Chavez Sentencing Transcript ("Chavez S.Tr.") at 9). But he argued that the total recommended sentence "would overstate the sentence the defendant should receive in light of the [18 U.S.C. §] 3553(a) factors" (id. at 7), including the factor of sentencing disparity among defendants (see id. at 8); counsel argued that the district court "does have discretion" and suggested "the 30-year sentence on the gun [count] itself . . . is more than sufficient to cover the elements of sentencing in this case" (id. at 7).
The district court's response to this suggestion was that a reduction of the sentence for count one on that theory would face a legal obstacle:
The statute, if I'm not mistaken, on which the Count 2 conviction rests requires that the sentence imposed on that count be consecutive to any sentence imposed on the underlying count. And I have some concern that it would violate the spirit, and possibly the letter of that statute, if I were to say, well, the sentence I might normally impose is . . . 20 years, for the narcotics count, but since he's getting 30 anyway, that's enough and I'll give him a year and a day or something on the narcotics count and then impose the 30-year count consecutive.
(Id.)
The district court opined that a total sentence of 720 months for Chavez (i.e., the 30-year term at the bottom of the Guidelines-recommended range for count one plus the 30-year consecutive term statutorily required for count two) would be "effectively a life sentence and more" (id. at 8), and that such a total would be an "extraordinary" sentence (id.) for "a 30-year-old man who has no actual violent crimes to his record" (id. at 17) and might not be "necessary to protect the public or to adequately punish him for his crimes" (id.). Nonetheless, the court concluded that it had "no doubt that a lengthy prison term" for count one itself "[wa]s required" (id. at 17), given that "Chavez [wa]s a major drug trafficker by anyone's definition" (id. at 16), that "[h]e was a significant leader in a conspiracy that distributed huge amounts of cocaine" (id.), that "he and the colleagues he supervised possessed numerous firearms" (id.), that "[t]here can be no doubt that he and those who worked for him were prepared to engage in acts of violence in furtherance of their drug distribution schemes" (id.), that he "has a criminal record starting as a juvenile, including a previous drug trafficking conviction and a previous conviction for possession of a firearm starting at a very young age" (id. at 17), and that he undisputedly "is a professional narcotics dealer with no respect for the law and a serious potential for violence" (id.). The court
conclude[d] that a 25-year sentence on the underlying narcotics conspiracy here is reasonable. The statutory mandatory minimum sentence here is 20 years, and that would apply even if the amount of cocaine involved in the conspiracy were vastly lower than what was shown in this case.
(Id. at 18.) The court noted that this sentence did not include any enhancement for Chavez's possession of the firearm and that had there been no firearm count, the sentence on the narcotics conspiracy count would have been higher. (See id. at 19-20); see generally Guidelines § 2D1.1(b)(1) (requiring a two-step increase in offense level if a firearm was possessed in connection with a drug trafficking crime); but see id. § 2K2.4(b) and Application Note 4 (If a *133 sentence for a § 924(c) offense "is imposed in conjunction with a sentence for an underlying offense, do not apply any specific offense characteristic for possession . . . [of the] firearm when determining the sentence for the underlying offense. . . . Do not apply any weapon enhancement in the guideline for the underlying offense. . . .").
The district court concluded that as a matter of law it could not properly reduce Chavez's count-one narcotics conspiracy sentence based on the length of the sentence required for count two because such a reduction would effectively violate the statutory requirement that the sentence on count two be consecutive:
[W]hat drives th[e] outcome, it seems to me, is the specific statute that specifically requires a 30-year sentence, and not just a 30-year sentence, but a consecutive 30-year sentence on the gun count. And that is a command of Congress.

(Chavez S.Tr. 8 (emphases added).)
[D]espite the discretion in sentencing granted by [United States v.] Booker[, 543 U.S. 220, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005),] and Section 3553(a), I have to respect the commands of Congress. Congress imposed the 30-year mandatory sentence, and Congress specifically required that the sentence be consecutive to whatever other sentence is imposed. So I don't think I can be faithful to the law by lowering the narcotics sentence because the total sentence, including the mandatory consecutive sentence, appears to be [ ] too high. Rather, I believe I must independently decide what is [ ] an appropriate sentence for the narcotics crime and then impose the mandatory consecutive sentence on top of that.

(Id. at 18 (emphases added).) Having decided that the appropriate prison term for Chavez's narcotics conspiracy conviction on count one was 25 years, the court imposed the required 30-year term consecutively for count two, making Chavez's total prison term 55 years, or 660 months.
In this appeal, Chavez's principal contention is that the district court erred in believing that, in arriving at a reasonable total sentence, it was not authorized to impose a shorter prison term for count one in light of the severe consecutive prison term it was required to impose on count two. We reject this contention.
Although this Court has not yet addressed this question, we dealt with a similar issue in United States v. Stanley, 928 F.2d 575 (2d Cir.1991) ("Stanley"), in the context of a departure from the then-mandatory Guidelines. In Stanley, the district court had been concerned about a sentencing disparity resulting from the exercise of prosecutorial discretion benefiting defendants who entered into plea agreements in which they admitted their guilt to drug trafficking charges. Those defendants were not charged with a firearm count under § 924(c); but for a defendant who chose to go to trial on similar drug trafficking charges, the government added a § 924(c) count that exposed that defendant to an additional mandatory prison term of five years or more. For the defendant in Stanley, who chose to go to trial and was charged with and convicted of both drug trafficking and firearm possession, the imprisonment range recommended by the Guidelines on the narcotics count was 87-108 months, although the recommended range would have been 108-135 months if there had been no § 924(c) count because a firearm enhancement would then have been required for the narcotics count. Stanley's conviction on the § 924(c) count required an additional sentence of five years. The district court departed downward from the Guidelines range and sentenced Stanley to 60 months on the narcotics count; the additional five-year sentence *134 on the firearm count brought Stanley's total prison term to 120 months, i.e., a term that would have been within the Guidelines range had there been no § 924(c) count.
We found that the government's election as to what charges to bring against Stanley was within the proper bounds of prosecutorial discretion, see Stanley, 928 F.2d at 578-79, and we thus concluded that the downward departure granted by the district court on the narcotics count was impermissible, stating in part as follows:
It was the intention of Congress that the five-year penalty mandated by § 924(c) be imposed "in addition to any other term of imprisonment." United States v. Lawrence, 928 F.2d 36, 38 (2d Cir. 1991) (emphasis added). By reducing the "other term of imprisonment," i.e., the sentence on the underlying narcotics offense, the district court ensured that the defendant would not receive any additional imprisonment term because of his § 924(c) conviction. Although technically complying with the statute, the sentence imposed nullified the legislative intent of additional punishment for violating § 924(c).
Stanley, 928 F.2d at 582 (emphases in Stanley).
This observation has equal force here. Section 924(c) provides severe penalties for any person convicted of possessing a firearm in furtherance of a federal drug trafficking crime, including a "term of imprisonment of not less than 30 years" if the weapon was "equipped with a firearm silencer." 18 U.S.C. § 924(c)(1)(B)(ii). The section was plainly designed to impose penalties that are cumulative to the penalties imposed for other crimes. First, it provides that these penalties "shall" be imposed "in addition to the punishment provided for [the underlying] drug trafficking crime." 18 U.S.C. § 924(c)(1)(A) (emphasis added). Second, § 924(c) provides that "[n]otwithstanding any other provision of law,"

no term of imprisonment imposed on a person under this subsection shall run concurrently with any other term of imprisonment imposed on the person, including any term of imprisonment imposed for the . . . drug trafficking crime during which the firearm was used, carried, or possessed.
Id. § 924(c)(1)(D) (emphasis added). Thus, Congress expressly provided that the prison term imposed for such a firearm offense must be consecutive to the term imposed for the underlying offense.
Section 3553(a) of 18 U.S.C. sets out factors that the sentencing court must consider in determining an appropriate sentence, and these factors have been accorded greater prominence in the wake of the ruling in United States v. Booker, 543 U.S. 220, 245, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), that the Guidelines are not mandatory. The § 3553(a) factors include the need for the sentence imposed to reflect the seriousness of the offense, to promote respect for the law, to provide just punishment for the offense, to afford adequate deterrence to criminal conduct, to protect the public from further crimes of the defendant, and to avoid unwarranted sentence disparities among defendants with similar records who have been found guilty of similar conduct; and consideration of only the factors set out in § 3553(a) could lead the court to conclude that a shorter total sentence than the total specified for a § 924(c) conviction and recommended for the underlying crime would be appropriate. As set out above, however, § 924(c) instructs that its mandatory minimum penalties are not to be made concurrent with any other penalties, including the penalty for the offense underlying the § 924(c) offense, "[n]otwithstanding any other provision *135 of law." 18 U.S.C. § 924(c)(1)(D). See also Kimbrough v. United States, ___ U.S. ___, 128 S.Ct. 558, 573, 169 L.Ed.2d 481 (2007) ("[S]entencing courts," although permitted by § 3553(a), after Booker, to deviate from an advisory-Guidelines-recommended range of imprisonment based on their policy disagreements with the Guidelines, "remain bound by the mandatory minimum sentences prescribed in the [statutes].").
Thus, as the Seventh Circuit stated in addressing an issue similar to that presented here, there is
tension with section 3553(a), but that very general statute cannot be understood to authorize courts to sentence below minimums specifically prescribed by Congress. . . . That was the rule when the guidelines were mandatory,. . . and it was not changed by Booker. For in making the sentencing guidelines advisory, the Court did not authorize courts to sentence below the minimums prescribed not by the guidelines but by constitutional federal statutes.
United States v. Roberson, 474 F.3d 432, 436 (7th Cir.2007); see id. at 434 (Booker "did not authorize district judges to ignore statutory sentencing ranges").
In sum, we conclude that a sentencing court is required to determine the appropriate prison term for the count to which the § 924(c) punishment is to be consecutive; and if the court reduces the prison term imposed for that underlying count on the ground that the total sentence is, in the court's view, too severe, the court conflates the two punishments and thwarts the will of Congress that the punishment imposed for violating § 924(c) be "addition[al]" and "no[t] . . . concurrent[ ]."
Our Sister Circuits that have addressed this question have similarly reached this conclusion. See, e.g., United States v. Hatcher, 501 F.3d 931, 933 (8th Cir.2007) (sentencing court could not permissibly "conflate[ ] the sentences for the § 924(c) offenses and the related [underlying] crimes"), cert. denied, ___ U.S. ___, 128 S.Ct. 1133, 169 L.Ed.2d 956 (2008); United States v. Franklin, 499 F.3d 578, 584-85 (6th Cir.2007) ("When any downward variance of the guideline range is based upon the effect of a mandatory sentence, congressional intent is repudiated, just as if the mandatory sentence itself had been reduced."); United States v. Roberson, 474 F.3d at 436 ("[T]o use the presence of a section 924(c)(1) add-on to reduce the defendant's sentence for the underlying crime would be inconsistent with Congress's determination to fix a minimum sentence for using a firearm in [the underlying crime]. If the judge reduces the defendant's sentence on the underlying crime . . . from, say, 50 to 49 months because the defendant [violated § 924(c)(1) and] must be sentenced to 84 months on top of the sentence for the underlying crime, the effect is to reduce the statutory minimum sentence from 84 months to 83 months." (second emphasis in original; other emphases added)).
We conclude that the district court in the present case correctly reasoned that it was required to determine an appropriate prison term for Chavez on the narcotics count and correctly concluded that it was not permitted to reduce that prison term on account of the mandatory minimum sentence provided by § 924(c) for the firearm count.

C. Other Sentencing Contentions


1. Chavez

Chavez also contends that the district court, in making its Guidelines calculations, erred in increasing his offense level on the grounds that he was a leader of the conspiracy and was responsible for conspiring *136 to distribute more than 150 kilograms of cocaine. These contentions are without merit and do not require extended discussion.
The Guidelines provide that the offense level of a defendant is to be increased if he "was an organizer or leader of a criminal activity that involved five or more participants or was otherwise extensive." Guidelines § 3B1.1(a). While the criminal activity must be found to have involved five or more participants, the defendant need not have been the leader of more than one other participant for this adjustment to apply. See id. Application Note 2; see also United States v. Zichettello, 208 F.3d 72, 107 (2d Cir.2000), cert. denied, 531 U.S. 1143, 121 S.Ct. 1077, 148 L.Ed.2d 954 (2001).
The record here shows that the conspiracy of which Chavez was convicted involved many more than five participants, including Chavez, Barraza who supervised the New York operation, Daniel Barraza, Jose Luis Barraza, Ibarra, Che and other distributors to whom Ibarra delivered cocaine, and Acosta. And the evidence discussed in Parts II.A.1. and 2. above showed that Chavez was a leader of the conspiracy, as he, inter alia, gave instructions to Barraza for the dilution of narcotics in Barraza's possession; had the power to fire Ibarra, who had been hired by Barraza; and instructed Barraza to hide in Mexico when law enforcement agents were closing in on the organization's operations. Chavez also acknowledged his own personal responsibility for the "debt" to cocaine suppliers that resulted from the government's seizure of large quantities of cocaine from coconspirators in New York. The district court correctly increased Chavez's offense level for a leadership role in the conspiracy.
With respect to narcotics quantity, the Guidelines provide that a defendant is accountable not only for all the narcotics with which he was directly involved, but also, "in the case of a jointly undertaken criminal activity, all reasonably foreseeable quantities of [narcotics] that were within the scope of the criminal activity that he jointly undertook." Guidelines § 1B1.3 Application Note 2. The record here was ample to show that the organization's New York operations in 2002 alone involved more than 150 kilograms of cocaine. It included Ibarra's testimony that his first task for Barraza in May included picking up 50 kilograms from Acosta and delivering them to the organization's distributors (see Tr. 397-400); that in September the quantities of cocaine that Ibarra got from Acosta on three occasions totaled 46 kilograms (see id. at 405-06, 432-34); and that on two occasions in September, Ibarra went to Connecticut and picked up a total of 140 kilograms of cocaine from another supplier for delivery to the organization's distributors (see id. at 441-42, 468-69, 471). Thus, in these instances alone, the operation overseen by Barraza handled 236 kilograms of cocaine. As Chavez was Barraza's boss, these quantities were foreseeable to Chavez, and the challenge to the district court's finding that he was responsible for more than 150 kilograms of cocaine is merit less.

2. Acosta

Acosta's conviction of the conspiracy charged in count one subjected him to a mandatory minimum prison term of 10 years. See 21 U.S.C. §§ 841(a), 841(b)(1)(A), 846. The district court calculated that the advisory-Guidelines-recommended range of imprisonment for him was 188-235 months. Acosta  age 50  urged the court to sentence him to less than 188 months, arguing principally (1) that he had had an unfortunate childhood, being abandoned by his father under the *137 age of 10, not knowing who his real family was, thinking his grandmother was his mother and then learning otherwise, and being taken away from even her, and (2) that even a 10-year term would mean that Acosta would be in prison until he was past the age of 60. The court sentenced Acosta to 198 months' imprisonment. On appeal, Acosta contends that the court erred because it rejected his contentions in the belief that it could not take his tragic childhood into account unless his experiences caused his crime, and that the court did not even consider his age. We disagree with Acosta's characterization of the court's decision.
In sentencing Acosta, the district court stated in part as follows:
Mr. Acosta . . . was a significant member of a very major drug ring. He had significant responsibility in handling large quantities of cocaine. I pause to note [Acosta's] unfortunate upbringing, and that certainly is tragic. And I feel for the child that [Acosta's counsel] described. At the same time, what stands before me is not a child but a man, a man who is 50 years old who has  and the nature of the crime here is not a crime of violence or a crime of passion or something that could be explained by psychological damage caused by an unfortunate childhood. What we have here is a crime carried out as a profession, as a deliberate choice for making money.
(Acosta Sentencing Transcript at 11.) We read these statements not as implying that the district court was imposing a requirement that the defendant's unfortunate upbringing be causally related to the crime at hand, but rather as showing that the district court considered Acosta's age and childhood and found that they did not constitute "mitigating circumstances" (id.) and that the circumstances of Acosta's crime and conduct weighed against a lower sentence.
In sum, the record reflects the district court's proper consideration of the § 3553(a) factors, and we see no basis for finding that Acosta's sentence was unreasonable.

CONCLUSION
We have considered all of the arguments made by Chavez and Acosta in support of their appeals and have found them to be without merit. The judgments of the district court are affirmed.
