CHIN, Circuit Judge:
On January 23, 2009, defendant-appellant Sonny Desposito set a car on fire at a residence in Chestnut Ridge, New York. Then, with the police distracted by the fire, he robbed the M
&
T Bank, stealing approximately $10,000. After his arrest, he sent a series of letters from jail seeking to persuade a relative and a friend to create falsified evidence he hoped would raise a reasonable doubt at trial. The letters were intercepted and turned over to the police. Desposito was convicted of bank robbery, “us[ing] fire ... to commit [a] felony,” and attempting to obstruct an official proceeding, in violation of 18 U.S.C. §§ 2113(a), (d), 844(h)(1), and 1512(c)(2). He appeals the convictions of using fire to commit a felony and attempting to obstruct an official proceeding. We affirm.
STATEMENT OF THE CASE
A.
The
Facts
1.
The Bank Robbery
On January 21, 2009, Desposito performed a test run of his scheme to rob the M & T Bank in Chestnut Ridge, New York. He began by setting fire to a car parked in a residential driveway in the vicinity and then practiced driving from that location to the bank. Because the car did not fully engulf in flames, the next day he arranged for his friend Betty Mastrar-rigo to buy him a can of lighter fluid from the A & P grocery store where she worked.
The following day, on January 23, Des-posito carried out his scheme. First, he set fire to another car parked in a residential driveway about two minutes away from the M & T Bank. Then, as police received 911 calls about the fire, which completely destroyed the car, Desposito proceeded to the bank. Wearing a black ski cap and carrying what appeared to be a pistol (but turned out to be a pellet gun), Desposito robbed the bank by threat of force and stole approximately $10,000.
He then escaped through the woods to his waiting car and drove to Pearl River, New York. Once there, he pulled over on a side street and burned the lighter fluid can and the clothes he had worn during the robbery in a pile on the sidewalk. After he sped away, a nearby resident found the fire and put it out. A black ski cap was found among the debris; testing later determined that the cap contained traces of Desposito’s DNA.
2.
The Attempt to Obstruct His Trial
Desposito was later arrested. While he was in jail awaiting trial, he wrote ten letters and mailed them in three sets to his friend Besnick Ljuljanaj, to whom Despo-sito had confessed his crime and in whose house he had hidden the stolen money.
Ljuljanaj’s father intercepted all of these letters when they were delivered and turned them over to his attorney, who then delivered them to law enforcement.
The first mailing, dated April 9, 2009, contained six letters. In one letter addressed to Ljuljanaj, Desposito asked him, among other things, to deliver the other five letters to the respective addressees. He also suggested that Ljuljanaj read the letter addressed to Desposito’s half-brother Kris Fortier so he would understand why “the next time I send you mail, it will have a bag with my prints on it to give to [K]ris.”
In Fortier’s letter, Desposito directed his half-brother to plant a new lighter fluid can that could be introduced into evidence at his criminal trial. As part of the plan, Fortier was supposed to send him some A & P grocery bags in a care package so that he could place his fingerprints on them. Desposito would then send the bags back to Ljuljanaj, who would forward them to Fortier. Once he received them, Fortier was supposed to place a new lighter fluid can, with Betty Mastrarrigo’s fingerprints on it, inside the bags so it would appear that Desposito had never used the original can of lighter fluid Mastrarrigo had bought for him. Desposito’s final instruction read: “Pat yourself on the back, you just rose reasonable doubt and saved my life.”
Another enclosed letter was addressed to Mastrarrigo. In it, Desposito tried to convince her to put her fingerprints on the new can of lighter fluid. Rather than tell her his plan, however, Desposito lied and told her that his father had found the original can, but had contaminated it with his own fingerprints. He asked her to touch the can again so a fingerprint analysis would verify that it was the same can.
On April 24, 2009, Desposito sent a second envelope containing three similar, but shorter, letters for Ljuljanaj, Fortier, and Mastrarrigo because he was afraid the first package had not reached them. In Ljuljanaj’s letter, Desposito asked him to check at the post office to see if the first package of letters was awaiting payment of additional postage because it was too heavy.
On May 1, 2009, Desposito sent a third envelope with a single letter for Ljuljanaj. In it, Desposito asked Ljuljanaj to write back to confirm that he was receiving his letters. He also told Ljuljanaj to disregard everything in the prior two letters if he had received them. Desposito apparently had not abandoned his plan, however, because police found five more letters addressed to Desposito’s father in the family residence. These letters contained messages for his half-brother Fortier, notifying him of the three mailings to Ljuljanaj and instructing him to find out whether Ljuljanaj had received them and, if not, to retrieve them from the post office.
B.
Proceedings Below
On August 12, 2010, Desposito was indicted for committing bank robbery in violation of 18 U.S.C. § 2113(a), (d), using fire to commit a felony in violation of 18 U.S.C. § 844(h)(1), and attempting to obstruct, influence, or impede an official proceeding in violation of 18 U.S.C. § 1512(c)(2).
On November 30, 2010, a jury trial commenced before Judge Seibel. Desposito took the stand in his own defense and intimated that Ljuljanaj had committed the bank robbery. When asked about the letters, Desposito admitted writing them, but explained that he “constructed them in such a way that [Ljuljanaj] could exonerate [Desposito] without directly implicating himself, without betraying his own involvement in any way.” After this testimony, the government asked for a sidebar and sought a ruling
in limine
on the admissibility of evidence about a second plan contained in Desposito’s letters.
The second plan related to a pending New Jersey state prosecution against Des-posito. The letters showed that Desposito was plotting to discourage the victim of the New Jersey crime from testifying by having Ljuljanaj impersonate someone from the Witness and Victim Advocacy Center. These portions of the letters had been redacted during the government’s case-in-chief.
Over the defendant’s objection, the district court ruled that such evidence was admissible to impeach Desposito’s character for truthfulness, pursuant to Federal Rules of Evidence 608(b) and 403. When the government began to ask Desposito about the second plan, the district court immediately instructed the jury about the limited purpose of the questioning. Ultimately, Desposito admitted asking Ljulja-naj to carry out the second plan, even though Ljuljanaj bore no responsibility for that crime.
On December 8, 2010, the district court delivered its charge to the jury. In instructing the jury on the crime of using fire to commit a felony, the court explained:
In order to prove that the defendant used fire, the government must prove beyond a reasonable doubt that the defendant actively employed fire to commit the felony. To use has its common meaning, to employ, to avail oneself of, or to carry out a purpose or action by means of. To prove that fire was used to commit a felony, the government must prove beyond a reasonable doubt that the fire was a means used by the defendant to carry out the felony. It must be an integral part of the felony, not something incidental or independent or that merely happened to facilitate or assist it. The fire must be part and parcel of the predicate crime.
So to satisfy this element, you must also find the defendant knowingly used fire, that means used fire purposely and voluntarily and not by accident or mistake.
The same day, the jury found Desposito guilty of all three counts. The district court subsequently sentenced Desposito to concurrent 108-month terms of imprisonment on the bank robbery and attempt to obstruct justice counts, and a consecutive
120-month term on the use of fire count, as required by 18 U.S.C. § 844(h)(1).
The judgment of conviction was filed on June 17, 2011. This appeal followed.
DISCUSSION
Desposito challenges his conviction on several grounds. First, he argues that the evidence was insufficient to convict him of using fire to commit a felony because it only showed that he used fire to
facilitate
the robbery, rather than to
commit
the crime. Second, even if § 844(h)(1) proscribes using fire as he did, Desposito argues that convicting him in this case violates his Due Process rights because he lacked fair warning that his conduct was illegal. Third, he argues that the letters he wrote from jail were an insufficient basis for convicting him of attempting to obstruct an official proceeding because his scheme depended on the voluntary cooperation of others and there was no evidence they ever agreed to assist him. Finally, Desposito argues that permitting the government to question him about the pending New Jersey prosecution was overly prejudicial and warrants a new trial. We review each of these claims in turn.
A.
Using Fire to Commit a Felony
1.
Applicable Law
We review
de novo
challenges to the sufficiency of the evidence underlying a criminal conviction.
See United States v. Mahaffy,
693 F.3d 113, 123 (2d Cir.2012). Section 844(h)(1) of the Criminal Code provides, in relevant part, that “[wjhoever ... uses fire or an explosive to commit any felony which may be prosecuted in a court of the United States ... shall, in addition to the punishment provided for such felony, be sentenced to imprisonment for 10 years.” 18 U.S.C. § 844(h)(1). We have not yet had occasion to construe the phrase “uses fire ... to commit any felony.”
In construing a statute, we begin with the plain language, giving all undefined terms their ordinary meaning.
See Schindler Elevator Corp. v. United States ex rel. Kirk,
— U.S. -, 131 S.Ct. 1885, 1891, 179 L.Ed.2d 825 (2011);
Smith v. United States,
508 U.S. 223, 228, 113 S.Ct. 2050, 124 L.Ed.2d 138 (1993);
23-34 94th St. Grocery Corp. v. N.Y.C. Bd. of Health,
685 F.3d 174, 182 (2d Cir.2012). Absent an ambiguity, our analysis also ends with the statutory language.
See Schindler Elevator Corp.,
131 S.Ct. at 1893;
Devine v. United States,
202 F.3d 547, 551 (2d Cir.2000). “[W]e must presume that the statute says what it means.”
Devine,
202 F.3d at 551. We will resort to legislative history and other tools of statutory interpretation only if we conclude that the text is ambiguous.
See Auburn Hous. Auth. v. Martinez,
277 F.3d 138, 143-44 (2d Cir.2002).
2.
Application
Because the statute does not define the word “use,” we supply it with its ordi
nary meaning. The verb “use” means “to put into action or service,” “to avail oneself of,” or “to carry out a purpose or action by means of.” Merriam-Webster’s Collegiate Dictionary 1378 (11th ed.2004);
accord
Black’s Law Dictionary 1541 (6th ed.1990);
see also Bailey v. United States,
516 U.S. 137, 145, 116 S.Ct. 501, 133 L.Ed.2d 472 (1995),
superseded by statute,
Act [t]o throttle criminal use of guns, Pub.L. No. 105-386, § 1,112 Stat. 3469 (1998) (amending 18 U.S.C. § 924(c));
United States v. Ruiz,
105 F.3d 1492, 1503-04 (1st Cir.1997) (interpreting 18 U.S.C. § 844(h)(1)). Because “use” has a variety of meanings, the surrounding context helps clarify which one Congress intended.
See Bailey,
516 U.S. at 143, 145, 116 S.Ct. 501. Here the word appears in the phrase “uses fire ... to commit any felony.” 18 U.S.C. § 844(h)(1). Thus, to “use” fire means the accused must have carried out the crime by means of fire.
Giving the word this plain meaning, it is obvious that Desposito “used” fire to commit bank robbery. Indeed, as the evidence showed, the use of fire was an integral part of Desposito’s scheme to rob the bank. He admitted to Ljuljanaj that his plan was to rob the bank by diverting the police with the car fire. Two days before the robbery, Desposito rehearsed the robbery by setting fire to a car and driving to the bank to gauge the travel time. The next day, he obtained some lighter fluid so the fire would be more imposing. On the day of the robbery, he set fire to another car to divert the police and drove to the M & T Bank just as they received the first 911 call about the fire. While the police were preoccupied with the fire he had set, Desposito robbed the bank and escaped with $10,000. This evidence was more than sufficient to convict him of using fire to commit the felony, or in other words, to convict him of carrying out the bank robbery by means of fire.
Desposito argues that the fire only
facilitated
his crime and, in reliance on
United States v. Hayward,
6 F.3d 1241 (7th Cir.1993),
overruled on other grounds, United States v. Colvin,
353 F.3d 569 (7th Cir.2003), he insists that mere facilitation is beyond the scope of § 844(h)(1). In
Hayward,
the defendants were convicted of using fire to commit a conspiracy against civil rights because they had burned crosses on the lawn of a white family who had entertained black house guests.
See Hayward,
6 F.3d at 1243-44. The Seventh Circuit held that the text of § 844(h)(1)
clearly applied to the defendants’ use of fire to burn the crosses.
See id.
at 1245-46. When the defendants insisted that this interpretation would lead to absurd results, such as adding ten years to a car thiefs sentence because he used a cigarette lighter as a light source while he picked the car’s lock, the court reasoned that “there is a significant difference between using fire to
commit
a felony and using fire to
facilitate
or
assist
in the commission of a felony.”
Id.
at 1246. Using fire to light the crosses was “an integral part of the threat or intimidation” at the heart of the conspiracy against civil rights, which could not be replicated by illuminating the crosses with another source of light.
Id.
at 1247. Thus, the Seventh Circuit concluded that “[t]he fire did not simply
facilitate
or
assist
[the defendants] in the commission of their crime.”
Id.
We conclude that
Hayward
is of no help to Desposito. First, the distinction drawn in
Hayward
between committing and facilitating a felony was at best dicta. Despite drawing this contrast, the court did not hold that using fire to merely “facilitate” a felony was beyond the scope of the statute.
See id.
at 1246^7 & n. 7. Rather, the court was simply concluding that the defendants’ facilitation argument failed because the use of fire in the circumstances before it was not mere facilitation. Thus,
Hayward
does not support Desposito’s contention that mere facilitation falls outside the statute’s plain meaning.
Second, even assuming
Hayward
stood for that proposition, Desposito’s actions differ significantly from using a cigarette lighter as a source of light or illumination. Here, fire was critical to his scheme and made the robbery possible.
See Abuelha-wa v. United States,
556 U.S. 816, 820, 129 S.Ct. 2102, 173 L.Ed.2d 982 (2009). Using a cigarette lighter as a light source is merely incidental to the actual crime; a flashlight or other non-criminal tool could serve the same purpose just as well. The fire from the cigarette lighter poses no threat. In contrast, Desposito’s fire posed a threat to life and property, one that
demanded the immediate attention of the police. This threat was so integral to Des-posito’s scheme that he even rehearsed it and decided to use lighter fluid because the practice fire was not menacing enough. Under these circumstances, the Seventh Circuit’s hypothetical is clearly distinguishable.
Finally, even assuming,
arguendo,
§ 844(h)(1) does not encompass using fire to “facilitate” a felony, the district court instructed the jury that mere facilitation was not sufficient. The jury still found Desposito guilty. We must affirm the verdict “if the evidence, when viewed in its totality and in the light most favorable to the government, would permit any rational jury to find the essential elements of the crime beyond a reasonable doubt.”
United States v. Geibel,
369 F.3d 682, 689 (2d Cir.2004). For the reasons stated above, the jury’s conclusion that Desposito used fire to commit the bank robbery, and not merely to facilitate it, was entirely reasonable.
Desposito also asks us to consider the legislative history behind the Anti-Arson Act of 1982, which added using “fire” to § 844(h)(1).
See
Pub.L. No. 97-298, § 2(b), 96 Stat. 1319 (1982). Because we find that Desposito’s use of fire falls within the plain language of the statute, we need not consider any other tools of statutory interpretation. Even if we did find the statute ambiguous, however, we note that the legislative history would support our construction of the statute. According to the House Report, Congress intended to eliminate the need to prove that an explosive was the source of a criminal fire and chose to do so by also criminalizing the use of “fire” to commit felonies.
See
H.R.Rep. No. 97-678, at 2 (1982),
reprinted in
1982 U.S.C.C.A.N. 2631, 2632. In doing so, Congress was well aware that fire could be used to commit a variety of different crimes, and not merely arson and insurance fraud as Desposito contends.
See id.
(“Fire is used extensively not only for the criminal purposes of extortion, terrorism and revenge, but to conceal other crimes such as homicide, and for fraud against insurance companies.”);
see also United States v. Fiore,
821 F.2d 127, 132 (2d Cir.1987) (explaining that fire was added to remove the source-based limitation imposed on the term “explosive,” which was originally intended “‘to include all situations in which an explosive or material with explosive capacity was used criminally’ ” (quoting 128 Cong. Ree. H4957 (daily ed. Aug. 2, 1982) (statement of Rep. Hughes))). Using fire to divert the police from a bank robbery is surely comparable to the uses of fire Congress specifically intended, such as “to conceal” a homicide. Thus, even if we considered the legislative history, it would not support Desposito’s narrow reading of the statute. Accordingly, we apply the statute as written and affirm the conviction.
B.
Due Process
We review challenges to a statute’s constitutionality
de novo. See United States v. Al Kassar,
660 F.3d 108, 129 (2d Cir.2011). Due process provides a criminal defendant with the right to “fair warning ... in language that the common world will understand, of what the law intends to do if a certain line is passed.”
McBoyle v. United States,
283 U.S. 25, 27, 51 S.Ct. 340, 75 L.Ed. 816 (1931) (Holmes, J.);
accord United States v. Cullen,
499 F.3d 157, 163 (2d Cir.2007). In determining whether the defendant’s due process rights were violated, we will consider “whether the statute, either standing alone or as construed by the courts, made it reasonably clear at the time of the charged conduct that defendant’s conduct was criminal.”
United States v. Lanier,
520 U.S. 259, 267, 117 S.Ct. 1219, 187 L.Ed.2d 432 (1997).
Because we hold that the plain language of § 844(h)(1) proscribes Desposito’s use of fire to commit bank robbery, we also hold that the statute itself provided him with “fair warning.” If the statutory language alone provides clear notice that certain conduct is illegal, Due Process is satisfied and the government may prosecute such activity without waiting for “every conceivable challenge to a law’s validity.”
United States v. Chestman,
947 F.2d 551, 564 (2d Cir.1991). Indeed, prior to Desposito’s challenge, the
government
had prosecuted and convicted others for similarly using fire or explosives to commit bank robbery.
See, e.g., United States v. Tucker,
253 Fed.Appx. 718, 719-20 (10th Cir.2007) (non-precedential order and judgment);
United States v. Ramsey,
No. 01-005-04, 2006 WL 328371, at *1-2 (E.D.Pa. Feb. 10, 2006). Because the application of § 844(h)(1) to defendant’s conduct was clear from the face of the statute, Desposito’s conviction does not offend Due Process.
C.
Attempt to Obstruct an Official Proceeding
We also review
de novo
the defendant’s challenge to the sufficiency of the evidence underlying his conviction for attempt to obstruct an official proceeding.
See Mahaffy,
693 F.3d at 123. To convict Desposito of an attempt, the government had to prove that he “had the intent to commit the object crime and ... engaged in conduct amounting to a substantial step towards its commission.”
United States v. Farhane,
634 F.3d 127, 145 (2d Cir.2011).
Here, the object crime was “corruptly ... obstructing], influencing], or impeding] any official proceeding.” 18 U.S.C. § 1512(c)(2). In construing the similar “Omnibus Clause” in 18 U.S.C. § 1503(a),
the Supreme Court has held that “ ‘a person is not sufficiently charged with obstructing or impeding the due administration of justice in a court unless it appears that he knew or had notice that justice was being administered in such court’ ” because, in the absence of such knowledge, that person “necessarily lacked the evil intent to obstruct.”
United States v. Aguilar,
515 U.S. 593, 599, 115 S.Ct. 2357, 132 L.Ed.2d 520 (1995) (quoting
Pettibone v. United States,
148 U.S. 197, 206, 13 S.Ct. 542, 37 L.Ed. 419 (1893)). Thus, to satisfy the element of intent, the government must show a “nexus” between the defendant’s act and the judicial proceedings; that is, there must be “a relationship in time, causation, or logic” such that the act has “the natural and probable effect of interfering with the due administration of justice.”
Id.
at 599-600, 115 S.Ct. 2357 (internal quotation marks omitted). Because of its similarity to § 1503(a), we have previously incorporated Aguilar’s “nexus requirement” into § 1512(c)(2).
See United States v. Reich,
479 F.3d 179, 185-86 (2d Cir.2007) (citing
Aguilar,
515
U.S. at 600, 115 S.Ct. 2357). Thus, to satisfy the first element of attempt in this case, the government had to show that Desposito’s letters had the natural and probable effect of obstructing his criminal trial.
To satisfy the second element, the government had to show that the defendant took a substantial step toward committing the crime that was “ ‘strongly corroborative of the firmness of the defendant’s criminal intent.’ ”
Farhane,
634 F.3d at 146-47 (quoting
United States v. Stallworth,
543 F.2d 1038, 1040 & n. 5 (2d Cir.1976)). By utilizing the “substantial step” formulation, we have effectively adopted the Model Penal Code’s more liberal approach to punishing attempt crimes, rather than the narrower common law approach that waited until the defendant was within dangerous proximity to completing the crime.
See id.
at 146 (citing
Stallworth,
543 F.2d at 1040-41). A “substantial step” must be “ ‘something more than mere preparation, yet may be less than the last act necessary before the actual commission of the substantive crime.’ ”
Id.
at 147 (quoting
United States v. Manley,
632 F.2d 978, 987 (2d Cir.1980)). Whether specific conduct constitutes a substantial step depends on “ ‘the particular facts of each case’ viewed in light of the crime charged.”
Id.
(internal quotation marks omitted) (quoting
United States v. Ivic,
700 F.2d 51, 66 (2d Cir.1983)).
a.
Nexus
The letters sufficiently demonstrate a nexus between Desposito’s actions and his criminal trial. Throughout these letters, Desposito indicated that the purpose of his plan was to create fraudulent evidence — grocery bags and a can of lighter fluid bearing planted fingerprints — that would be admitted into evidence to raise a reasonable doubt as to his guilt. In his initial letter to Fortier, postmarked April 9, Desposito explained that he wanted to plant the fake lighter fluid can “so later on in my trial we can enter [it] into evidence and show the can that was found ain’t the one I bought. Ha ha!” Desposito then included a “to-do made easy list” for Fortier, in which the last direction was: “Pat yourself on the back, you just rose reasonable doubt and saved my life.” In his second letter, postmarked April 24, he again repeated: “[I]f you follow my instructions to a T, you’ll raise enough reasonable doubt to save my life.” Thus, the letters clearly demonstrate Desposito’s intent that the fabricated evidence would influence his criminal trial.
Desposito argues that the evidence fails to establish the required nexus because his plan relied on the voluntary cooperation of others and thus obstruction was not the “natural and probable effect” of his letters.
See Reich,
479 F.3d at 185. This argument is flawed because it focuses on factual causation and the likelihood that Despo-sito’s plan would succeed, even though it is well-established that success is not required to show the required nexus.
See Aguilar,
515 U.S. at 599, 115 S.Ct. 2357. As a component of the intent element, the proper inquiry is whether the defendant knew his actions would result in the obstruction of a specific judicial proceeding.
See id.
at 601, 115 S.Ct. 2357;
Reich,
479 F.3d at 185-86;
United States v. Quat-trone,
441 F.3d 153, 171 (2d Cir.2006) (“[T]his rule limits criminal liability to cases where the defendant has notice that his wrongful conduct will affect the administration of justice_”). Thus we have found the nexus requirement satisfied in situations where “the discretionary actions of a third person are required to obstruct the judicial proceeding” if it was “foresee
able to [the defendant] that the third party ... would act on the [communication] in such a way as to obstruct the judicial proceeding.”
Reich,
479 F.8d at 185.
Here, there was abundant evidence from which the jury could infer that Desposito intended and believed that the recipients of his letters would follow his instructions and obstruct his criminal trial. When Des-posito wrote to Ljuljanaj, “I know it’s asking ... a lot, but you would be saving my life,” the jury could reasonably infer that Desposito expected Ljuljanaj to do what he asked because Desposito advised Fortier in another letter that “you can trust [Ljuljanaj] with your life.” That Desposito had previously confided in Ljuljanaj about his crime and stored the stolen money in his bedroom corroborates this inference. Thus, a jury could reasonably conclude that it was foreseeable to Desposito that Ljuljanaj would help him when he sent the letters.
Even if the letters suggest that Desposi-to had some doubts about Ljuljanaj’s cooperation, they reveal no such reservations about Fortier, his half-brother. The letters to Fortier have a clearly different tone than those to Ljuljanaj, which include disclaimers such as:
I [am] sorry to put this much on you, and I’ll have no choice but to understand, if you don’t do any of this [stuff] for me.... You’re a great friend and a no to everything I’m asking of you can’t change that.
In contrast, Desposito’s letters to Fortier dispense with such niceties, beginning: “I got to keep this short and sweet because I need you to focus on saving my life.” They then proceed to list directions in the imperative voice, along with words of encouragement such as: “You got this. I know you do.”
These letters demonstrate that Desposi-to believed Fortier would comply upon receiving his letters. Because the letter to Fortier in the first mailing instructed him to send Desposito a care package containing A & P grocery bags, Desposito advised Ljuljanaj in an accompanying letter that “the next time I send you mail, it will have a bag with my prints on it to give to [K]ris.” When he did not receive the bags, Desposito apparently assumed Fortier never received his instructions as he sent nine more letters directing either Ljuljanaj to deliver the instructions to Fortier or Fortier to retrieve them. Even after Des-posito wrote in his third letter to Ljuljanaj to “just disregard everything,” he still believed Fortier would carry out the plan because he sent at least one more letter to his father, containing a message for Fortier to “contact Besni[c]k, see if he’s gotten any of the three letters I sent him.” Based on this evidence, a jury could conclude beyond a reasonable doubt that Des-posito knew the natural and probable consequence of sending his letters was that Fortier would create false evidence that would obstruct his criminal trial. Thus, the nexus requirement was satisfied.
b.
Substantial Step
Desposito argues that, under our precedent, merely communicating a plan to others cannot constitute a “substantial step.” Desposito points primarily to our holding in
United States v. Delvecchio,
816 F.2d 859 (2d Cir.1987), that “a verbal agreement alone, without more, is insufficient as a matter of law to support an attempt conviction.”
Id.
at 862. Based on this holding, Desposito argues that he cannot be convicted unless the government shows both that the letter recipients agreed to carry out the plan and that he then took an additional “substantial step” beyond that agreement.
Desposito takes our holding in
Delvec-chio
out of context. There are no bright
line rules for determining what actions amount to a “substantial step” and thus no requirement that the government show an agreement or any other specific circumstance.
See Farhane,
634 F.3d at 147;
see also Delvecchio,
816 F.2d at 861 (“Although the verbal formula for what constitutes a substantial step is clear, courts have not always found it easy to decide whether a defendant’s conduct has crossed over the line from ‘preparation’ to ‘attempt.’”). The analysis depends on the crime charged and whether the act tended to cause that particular crime to occur.
See Farhane,
634 F.3d at 147. Thus, as we have previously clarified, the agreement to purchase heroin in
Delvecchio
did not amount to a substantial step towards
possessing
heroin only because there was no “act to
effect possession,
such as acquisition, or attempted acquisition, of the purchase money, or travel to the agreed-on purchase site.”
Id.
(emphasis added).
Thus, our inquiry here turns on whether Desposito took any substantial steps to effect the obstruction of his criminal trial. The evidence clearly established that he did. First, Desposito wrote fifteen letters, ten sent to Ljuljanaj and five sent to his family, to implement his scheme to obstruct justice. In these letters, Desposito not only set out directions for Fortier and Ljuljanaj to falsify evidence, but he attempted to deceive Mastrarrigo into putting her fingerprints on the new can of lighter fluid. Second, Desposito mailed these letters from jail in at least four sets of mailings. When nothing resulted from his initial mailing to Ljuljanaj, he sent a second, lighter envelope, believing the first package had never been delivered because it was too heavy. When nothing came of the second mailing, Desposito changed course by sending another letter to Ljulja-naj, telling him to disregard his prior letters, and sending five additional letters directly to his family, instructing Fortier to retrieve the plans from Ljuljanaj. As a whole, these actions demonstrate that Des-posito tried to do everything he possibly could from jail to bring his plan to fruition and effect the obstruction of his trial. Thus, a rational jury could find beyond a reasonable doubt that his persistent writing and mailing of letters constituted substantial steps toward obstructing his criminal trial. As the evidence demonstrated both a nexus with the criminal trial and substantial steps, we affirm Desposito’s conviction for attempting to obstruct an official proceeding.
D.
Admissibility of Evidence
Finally, we review the district court’s evidentiary ruling for abuse of discretion.
See United States v. Cadet,
664 F.3d 27, 32 (2d Cir.2011). Under Rule 608(b) of the Federal Rules of Evidence, during cross-examination, district courts may permit questioning about “specific instances of a witness’s conduct ... if they are probative of the [witness’s] character for truthfulness or untruthfulness.” Fed. R.Evid. 608(b)(1);
see Lewis v. Baker,
526 F.2d 470, 475 (2d Cir.1975). Even if the specific instance is probative of the witness’s veracity, the court still must balance that probative value against the risk of unfair prejudice to the defendant, pursuant to Rule 403.
See
Fed.R.Evid. 608 advisory committee’s note;
United States v. Weic-
hert,
783 F.2d 23, 25 (2d Cir.1986). Under Rule 403, the evidence may be excluded if its prejudicial effect substantially outweighs its probative value.
See
Fed. R.Evid. 403. We afford great deference to the district court’s balancing under Rule 403,
see United States v. Contorinis,
692 F.3d 136, 144 (2d Cir.2012), and we will disturb it only if we find the decision to be arbitrary and irrational,
see United States v. Mercado,
573 F.3d 138, 141 (2d Cir.2009).
Although there was a risk the jury would consider Desposito’s pending New Jersey state criminal trial as evidence of his criminal propensity, we cannot say that the district court abused its discretion in concluding that this risk did not substantially outweigh the probative value of the evidence in light of Desposito’s testimony during his direct examination. When a defendant offers an exculpatory explanation for the government’s evidence, he “opens the door” to impeachment of his credibility, even by previously inadmissible evidence.
See United States v. Elfgeeh,
515 F.3d 100, 128 (2d Cir.2008). Because the plan to dissuade a witness from testifying in the New Jersey prosecution severely undermined Desposito’s innocent explanation for the plan to fabricate evidence in this case, it became highly probative of his credibility after his testimony on direct examination. At the same time, any risk of prejudice was mitigated because the district court issued a timely and appropriate limiting instruction and the government never revealed what the underlying New Jersey crime was.
See United States v. Williams,
205 F.3d 23, 34 (2d Cir.2000) (finding no undue prejudice where the other criminal charges are less serious than the charged crime and the district court issued a proper limiting instruction);
United States v. Livoti,
196 F.3d 322, 326 (2d Cir.1999) (same). We cannot say that the district court’s Rule 403 balancing was arbitrary or irrational. Therefore, the district court did not abuse its discretion by permitting impeachment on this topic.
CONCLUSION
We conclude that the evidence was sufficient to prove that Desposito used fire to commit the bank robbery, and we are satisfied that his conviction did not offend Due Process. We also conclude there was sufficient evidence to support his conviction for attempting to obstruct an official proceeding. Finally, the district court did not abuse its discretion by allowing the government to cross-examine Desposito about his plan to obstruct his New Jersey state criminal trial. Accordingly, the judgment of conviction is AFFIRMED.