JOSÉ A. CABRANES, Circuit Judge:

The Hague Convention on the Civil Aspects of International Child Abduction (the "Hague Convention" or "Convention"), opened for signature Oct. 25, 1980, T.I.A.S. No. 11,670, and the International Child Abduction Remedies Act ("ICARA"), 42 U.S.C. § 11601 et seq.,1 generally require courts in the United States to order children returned to their countries of "habitual residence" if those children have been wrongfully removed to, or retained in, the United States. See Chafin v. Chafin, ___ U.S. ___, 133 S.Ct. 1017, 1021, 185 L.Ed.2d 1 (2013). This appeal asks whether a child was "habitually resident" in Italy within the meaning of the Hague Convention and ICARA. As we explain, the Hague Convention uses the terms "habitual residence" and "habitually resident" in a practical way, referring to the country where a child usually or customarily lives. The term is not equivalent to the American legal concept of "domicile," which relies principally on intent. See note 5, post. Nonetheless, when parents move temporarily to another country, without agreeing to change the child's principal place of residence, a petitioner claiming that the new location has become the child's "habitual residence" must show that the child has "acclimated" to that country.2 Gitter v. Gitter, 396 F.3d 124, 134 (2d Cir.2005). The petitioner in this case failed to show that the parents agreed to settle in Italy, and he did not attempt to show that the child had acclimated there. Accordingly, the United States District Court for the Southern District of New York (Richard J. Sullivan, Judge) properly denied the petition for return of the child.

BACKGROUND

i.

Petitioner-appellant Gerardo Guzzo (the "Father") is an Italian citizen and resident of Scario, Italy, and respondent-appellee Luisa Maria Cristofano (the "Mother") is a United States citizen and resident of New York. They met in September 2005 onboard a flight from New York to Italy and began visiting each other regularly and discussing the prospect of marriage. In January 2006, the Mother discovered that she was pregnant. She soon visited the Father in Italy, where they resumed discussions about whether to get married in New York or Italy. The Father and Mother eventually agreed to marry in New York while maintaining their respective residences in Italy and New York.

In September 2006, their child was born. Based on the evidence presented at trial, the District Court found that from 2006 to 2007 the parties maintained their "bi-continental marriage," each parent visiting the other on numerous occasions, and in December 2007 the parties agreed that the Mother and the child would live primarily in Italy with the Father but return periodically to New York. During this time, however, the parents' relationship became increasingly tumultuous. In February 2009, the Mother took the child to New York and told the Father that she wanted a separation.

Over the next few months, the parents negotiated a separation agreement (the "Separation Agreement"), which the Mother signed in English on May 20, 2009, and which the Father signed in Italian on June 10, 2009. As relevant here, the Separation Agreement provided that the parents would "`continue to live separate and apart,'" that the Mother would "`have custody[] of the minor child,'" and that the child would attend school at the Good Counsel Academy in White Plains, New York. Guzzo v. Cristofano, No. 11 Civ. 7394(RJS), 2011 WL 6934108, at *2 (S.D.N.Y. Dec. 30, 2011) (quoting the Separation Agreement). The Separation Agreement also established a visitation schedule, which provided that the child would spend at least two months each year in Italy with the Father.

Soon after signing the Separation Agreement, the Mother returned to Italy with the child. As the District Court explained, the Mother "testified that her trip to Italy was undertaken as an attempt at reconciliation with [the Father], but that she was only willing to make the attempt because she had the protection of the Separation Agreement." Id. at *3. The Mother also testified that "regardless of the reconciliation attempt, she never intended to have the child attend primary school in Italy and that she always planned to live with the child in New York once he was in kindergarten." Id. With only intermittent vacations, including several trips to New York, the child continued to live in Italy, where he attended nursery school.

In November 2010, the Mother took the child to New York with the intention of not returning to Italy. The following month, the parents agreed to make another attempt at reconciliation, and the Mother moved back to Italy in January 2011 with the child. The effort was unsuccessful. In August 2011, the Mother returned with the child to New York, where they have lived ever since. The Mother also initiated divorce proceedings in Westchester County, New York.

ii.

The Father initiated this action under the Hague Convention in October 2011, alleging that the Mother had wrongfully removed the child from Italy in August 2011. The District Court held a three-day bench trial in November 2011. The next month, the District Court denied the petition, concluding that the Father had not proved by a preponderance of the evidence that Italy, rather than the United States, was the child's country of "habitual residence." Guzzo, 2011 WL 6934108, at *4.

The District Court began by restating our two-part test for determining a child's habitual residence. Id. at *5 (citing Gitter, 396 F.3d at 134). Under that test, a court must first "inquire into the shared intent of those entitled to fix the child's residence... at the [last] time that their intent was shared." Gitter, 396 F.3d at 134. If a court concludes that the parents did not intend to change a child's habitual residence, it then must assess "whether the evidence unequivocally points to the conclusion" that the child has acclimated to the new location, notwithstanding the parents' intentions. Id. The District Court noted that the Father had rested his petition entirely on the first prong, arguing that the parents had agreed to change the child's habitual residence to Italy; he had explicitly abandoned any argument that the child had acclimated to life in Italy. See Guzzo, 2011 WL 6934108, at *5 n. 2.

Based on the evidence presented at trial, the District Court determined that the parents' Settlement Agreement in 2009 exhibited their last shared intent regarding the child's usual residence. Although the Mother had moved to Italy with the child after signing the agreement, and had attempted to reconcile with the Father, the Court found "no evidence that the attempted reconciliation, in and of itself, altered the [Settlement] [A]greement in any way." Id. at *6. The Court also found the Father's testimony that the parents reconciled in June 2009 and formed a new shared opinion that the child would live in Italy to be "not credible." Id. at *7. As the Court explained:

Id. The Court further concluded that "[d]espite the parties' apparently sincere attempts at reconciliation, the evidence demonstrates that [the Mother] never contemplated spending her life in Italy or having the child attend Italian schools following preschool." Id.

With respect to the child's attendance at nursery school in Italy, the Court found that the Mother had "testified credibly that, regardless of the outcome of the attempted reconciliation, she intended to send the child to kindergarten in New York." Id. For instance, the Mother home-schooled the child in English "in order to prepare him for school in the United States." Id. She also retained ties to New York, by keeping her property there, and by "continu[ing] to perform work for New York clients and [to] collaborate with attorneys in New York." Id. at *8. The Mother also refused to register her marriage with Italian authorities, which would have enabled her to obtain Italian public health insurance. And the child "did not have Italian medical insurance, but rather was insured through Medicaid and received his primary medical treatment in the United States." Id. After reviewing the relevant case law, the Court concluded that "the evidence overwhelmingly demonstrates that, following the execution of the Separation Agreement, the parties never shared an intention to make Italy the child's habitual residence." Id. at *9. Accordingly, the District Court held that the Father had not established that the child's habitual residence was Italy, and denied the petition for return of the child.

This appeal followed.

DISCUSSION

A. Legal Framework

i.

"The Hague Convention on the Civil Aspects of International Child Abduction generally requires courts in the United States to order children returned to their countries of habitual residence, if the courts find that the children have been wrongfully removed to or retained in the United States." Chafin, 133 S.Ct. at 1021 (emphasis supplied). The question presented in this appeal is whether the Father met his burden of showing that the "habitual residence" of the child was Italy.

"Habitual residence," as one court has observed, "is the central — often outcome-determinative — concept on which the entire [Hague Convention] system is founded." Mozes v. Mozes, 239 F.3d 1067, 1072 (9th Cir.2001). Indeed, the terms "habitual residence" and "habitually resident" are used throughout the Hague Convention, see Gitter, 396 F.3d at 130 & n. 6 (listing uses), and determining the child's country of habitual residence is a threshold issue in nearly all Hague Convention cases, because the laws of that country determine the custody rights recognized by the Convention, see Hague Convention, art. 3.3

Despite the importance of the "habitual residence" concept, the drafters of the Hague Convention deliberately left that term undefined. See Holder v. Holder, 392 F.3d 1009, 1015 (9th Cir.2004). The official Hague Conference reporter described the term as "a well-established concept in the Hague Conference, which regards it as a question of pure fact, differing in that respect from domicile." Elisa Pérez-Vera, Explanatory Report: Hague Conference on Private International Law ¶ 66, in 3 ACTS AND DOCUMENTS OF THE FOURTEENTH SESSION 426, 445 (1982).4 Courts have widely recognized that the term should therefore be interpreted "according to `the ordinary and natural meaning of the two words it contains, as ... decided by reference to all the circumstances of any particular case.'" Koch v. Koch, 450 F.3d 703, 712 (7th Cir.2006) (quoting Mozes, 239 F.3d at 1071 (alterations omitted)).

Understood in an ordinary and nontechnical way, a child's "habitual residence" is simply the place where he usually or customary lives.5 See, e.g., Rydder v. Rydder, 49 F.3d 369, 373 (8th Cir.1995) (noting "no real distinction between habitual and ordinary residence" (citing Friedrich v. Friedrich, 983 F.2d 1396, 1401 (6th Cir.1993), in turn citing Re Bates, No. CA 122.89, High Court of Justice, United Kingdom (1989))). "`[I]n their natural and ordinary meaning[,] the words mean that the person must be habitually and normally resident [in that country], apart from temporary or occasional absences of long or short duration.'" Mozes, 239 F.3d at 1073 (quoting Shah v. Barnet London Borough Council, [1983] 1 All E.R. 226, 233 (Eng. H.L.)); see also id. at 1073 n. 13. Or, put another way, "we might say that if we observe someone centering his life around a particular location during a given period, so that every time he goes away from it he also comes back, we will call this his habitual residence." Id. at 1073-74. Under the Hague Convention, a petitioner bears the burden of establishing by a preponderance of the evidence a child's habitual residence at the time of the contested removal. See Mota v. Castillo, 692 F.3d 108, 112 (2d Cir.2012) (citing 42 U.S.C. § 11603(e)(1)(A)).

Although the concept of "habitual residence" seems straightforward, the issue is often disputed in Hague Convention cases, which tend to involve children with ties to more than one country. See Gitter, 396 F.3d at 133. Given the Congressionally recognized "need for uniform international interpretation of the Convention," 42 U.S.C. § 11601(b)(3)(B), courts have articulated guiding principles to provide for "intelligibility and consistency in [the] application" of this term,6 Mozes, 239 F.3d at 1072.

Determining a child's habitual residence often becomes difficult when parents move a child from one country to another, raising the question of which country is the "habitual residence" of the child. In evaluating these cases, we have looked to the intent of the parents as a particularly important factor in understanding the context of a child's move to another country. As we have explained:

Gitter, 396 F.3d at 132 (quoting Mozes, 239 F.3d at 1074). Accordingly, "we will presume that a child's habitual residence is consistent with the intentions of those entitled to fix the child's residence at the time those intentions were mutually shared." Id. at 133.

At the same time, however, "parental intent cannot alone establish a child's habitual residence," nor can it prevent a habitual residence from changing. Id. Instead, a child's habitual residence changes when the child becomes settled in another country, even if one or both of the parents intend for the child to return to the original country of habitual residence. Id. at 133-34. As we have explained:

Id. at 134 (some internal quotation marks omitted) (quoting Mozes, 239 F.3d at 1081). In other words, we ask: "[W]ould returning the children ... be tantamount to sending them home?" Holder, 392 F.3d at 1019. Accordingly, although "[n]ormally the shared intent of the parents should control the habitual residence of the child," that intent is not controlling when "the evidence unequivocally points to the conclusion that the child has [acclimated] to the new location." Gitter, 396 F.3d at 134. The Hague Convention, after all, "places at the head of its objectives the restoration of the status quo, by means of the prompt return of children" to their home country. Gitter, 396 F.3d at 130 (internal quotation marks omitted).

The two-step framework articulated in Mozes and followed by our Court in Gitter is flexible enough to account for the varied circumstances of individual cases. See, e.g., Nicolson v. Pappalardo, 605 F.3d 100, 105 (1st Cir.2010) (noting that the "habitual residence" inquiry does not amount to "rigid" adherence to parental intent); Koch, 450 F.3d at 715-16 (explaining that the framework in Mozes is "flexible" and "not so rigid"). When applying this framework, the age of the child and the time spent in the respective countries can affect how much weight a court should place on parental intent.7 For instance, parental intentions become less relevant the longer a child remains in the new environment. In fact, once a child has "been living in one country ... for a sufficiently long period," then "questions as to the purpose of the residence become irrelevant," Gitter, 396 F.3d at 134 (quotation marks omitted). Accordingly, although it makes sense to "`regard the intentions of the parents as affecting the length of time necessary for a child to become habitually resident, because the child's knowledge of these intentions is likely to color its attitude toward the contacts it is making,'" Ruiz v. Tenorio, 392 F.3d 1247, 1254 (11th Cir.2004) (quoting Mozes, 239 F.3d at 1079-80), courts must not forget that the core concern of "habitual residence" is where a child normally or usually lives. Once a court "`can say with confidence'" that the child has become settled into a new environment, habitual residence in that country is established. Gitter, 396 F.3d at 134 (quoting Mozes, 239 F.3d at 1081). In this way, the Gitter framework helps provide "intelligibility and consistency" to the habitual residence inquiry, Mozes, 239 F.3d at 1072, but it is flexible enough to recognize that the overall assessment of habitual residence is "not formulaic" but instead "is a fact-intensive determination that necessarily varies with the circumstances of each case,"8 Whiting v. Krassner, 391 F.3d 540, 546 (3d Cir. 2004).

In sum, "[t]o determine which country is a child's country of habitual residence under the Hague Convention, we apply the two-part test set forth in Gitter v. Gitter." Hofmann v. Sender, 716 F.3d 282, 291 (2d Cir.2013). We "begin an analysis of a child's habitual residence by considering the relevant intentions," because "[f]ocusing on intentions gives contour to the objective, factual circumstances surrounding the child's presence in a given location." Gitter, 396 F.3d at 132. We "presume that a child's habitual residence is consistent with the intention of those entitled to fix the child's residence at the time those intentions were mutually shared." Id. at 133. This presumption can be overcome, however, if the evidence shows that a child is settled into (or, "acclimated" to) the new environment — a burden that is more easily satisfied the longer a child has lived in that country. When considering these two steps, the court must not lose sight of the fact that the framework is designed simply to ascertain where a child usually or customarily lives.

ii.

On appeal, we review a district court's factual findings for clear error and its legal conclusions de novo. See Ozaltin, 708 F.3d at 368. The habitual residence inquiry is heavily fact dependent, but whether the relevant facts satisfy the legal standard is a question of law that we review de novo. See Gitter, 396 F.3d at 133 n. 8; see also Ozaltin, 708 F.3d at 368 ("Legal conclusions include interpretations of the Convention and applications of the appropriate legal standards to the facts."). Accordingly, "a determination of habitual residence under Article 3 of the Hague Convention is a mixed question of law and fact, under which we review essentially factual questions for clear error and the ultimate issue of habitual residence de novo." In re B.Del C.S.B., 559 F.3d 999, 1008 (9th Cir.2009) (internal quotation marks omitted); see also, e.g., Larbie v. Larbie, 690 F.3d 295, 312 (5th Cir.2012) (citing Barzilay v. Barzilay, 600 F.3d 912, 916 (8th Cir.2010)); Maxwell v. Maxwell, 588 F.3d 245, 250 (4th Cir.2009). We conclude that a district court "clearly erred" only if a review of the record "leave[s] us with `the definite and firm conviction that a mistake has been committed.'" Hofmann, 716 F.3d at 292 (quoting Mota, 692 F.3d at 114).

B. Analysis

i.

The Father argues, in part, that we need not assess whether the child's habitual residence changed to Italy because the child was habitually resident in Italy throughout his whole life. Accordingly, we begin by examining the District Court's conclusion that the child was habitually resident in the United States in 2009, at the time of the parents' purported last shared intent regarding the child's residence.

Even assuming that the child was habitually resident in Italy prior to 2009, we conclude that his habitual residence changed to the United States after the parties reached the Settlement Agreement in May and June of 2009. As noted, habitual residence depends on a combination of parental intent and physical presence, see, e.g., Mozes, 239 F.3d at 1079, but when a child is very young, "the shared intent of the parents ... [is] of paramount importance," Whiting, 391 F.3d at 550. We detect no clear error in the District Court's finding that the Separation Agreement — signed by the Mother and Father in May 2009 and June 2009, respectively — demonstrated the parents' shared intent for the child to live primarily in New York. Moreover, the child, who was then less than three years old, had been living with the Mother in New York for several months. Because the child was living in New York, and because the parents agreed that he would continue to reside in New York, we agree that the child, in light of his age, was habitually resident in the United States at the time of the Settlement Agreement.

The Father argues that the Settlement Agreement is invalid under Italian and New York law and therefore cannot support the District Court's finding of shared intent. We disagree. "Regardless of whether the document is enforceable in [state] court, it is nevertheless clearly probative of the parties `last shared intent' for the purposes of determining habitual residence under ICARA." Guzzo, 2011 WL 6934108, at *6 n. 3 (quoting Gitter, 396 F.3d at 134). Indeed, the Father acknowledged at trial that when he signed the Agreement he understood (1) its terms; (2) that it provided for the child's residence in New York; and (3) that it would be legally binding, even though he hoped to reconcile with the Mother. Id. at *5-6. Accordingly, we find no clear error in the District Court's finding regarding the parents' shared intent that the child would reside in New York. See Mozes, 239 F.3d at 1076 (noting that shared intent can exist "despite the fact that one parent may have had qualms" about the agreement).

ii.

Having found no error in the District Court's determination that the child was habitually resident in the United States at the time of the Settlement Agreement in 2009, we now consider whether the child became habitually resident in Italy following his return to that country in the summer of 2009.

At the first step of the Gitter test, we ask whether the evidence offered at trial showed "settled mutual intent from which... abandonment [of the prior habitual residence] can be inferred." Mozes, 239 F.3d at 1077. When considering this issue, "the court should look ... at actions as well as declarations." Gitter, 396 F.3d at 134. "Clearly, this is one of those questions of historical and narrative facts in which the findings of the district court are entitled to great deference." Mozes, 239 F.3d at 1077-78 (internal quotation marks omitted); see also Hofmann, 716 F.3d at 292 ("The last shared intent of the parents is a question of fact, and the district court's determination in that regard is reviewed for clear error and thus entitled to deference.").

Having reviewed the record and the parties' submissions, we conclude that the District Court's findings are "amply supported by the record, and there is nothing... leav[ing] us with `the definite and firm conviction that a mistake has been committed.'" Hofmann, 716 F.3d at 292 (quoting Mota, 692 F.3d at 114). Although the Mother agreed to the child's return to Italy and attendance at an Italian nursery school, the District Court found credible the Mother's testimony that her stay in Italy was temporary, and that she consistently intended to return to New York for the child to begin kindergarten. Record evidence amply supports this conclusion. For instance, the Mother and the child entered Italy on temporary tourist visas, and they registered for health care in New York even though eligible for public health insurance in Italy. The District Court also credited the Mother's testimony that "her willingness to attempt a reconciliation in Italy was clearly premised on the understanding that, should the reconciliation prove unsuccessful, the parties would continue to abide by the terms of the agreement." Guzzo, 2011 WL 6934108, at *9. Indeed, the Mother returned to New York with the child in November 2010, with the stated expectation of staying permanently in the United States, before she agreed to make another attempt at reconciliation with the Father the following month. Accordingly, we do not disturb the District Court's finding that the parents never shared an intent for their child to abandon his prior habitual residence in the United States.

iii.

The second step in the Gitter framework is to examine whether, notwithstanding a lack of shared parental intent to change the child's long-term residence, the child was nonetheless sufficiently settled into (or, "acclimated" to) the new environment such that returning the child to that environment would "be tantamount to sending [him] home." Holder, 392 F.3d at 1019.

In this case, the five-year-old child lived mostly in Italy from soon after his birth in 2006 until his removal in 2011, and he regularly attended nursery school there. If it were properly raised in this appeal, we might conclude that the child was "acclimated" to living in Italy — that is, we might be able to "`say with confidence'" that the child's usual or customary place of residence was Italy, notwithstanding any parental intentions to the contrary. Gitter, 396 F.3d at 134 (quoting Mozes, 239 F.3d at 1081). Compare, e.g., Shalit v. Coppe, 182 F.3d 1124, 1128 n. 5 (9th Cir. 1999) (noting, without deciding habitual residence, that "[t]hree years is certainly enough time for [the child] to be considered `settled' in Israel, regardless of [the mother's] claimed intention to have him return permanently to Alaska at some point in the future."), with Hofmann, 716 F.3d at 294 (no acclimation to New York where child had lived there for "over a year" but had moved within the state and had maintained "close ties to friends and caretakers in Canada").

We need not address this issue, however, because the Father — represented by counsel in the District Court and at the time of filing his opening appellate brief — did not preserve any argument that the child was acclimated to Italy. See Guzzo, 2011 WL 6934108, at *5 n. 2. "In our adversary system, in both civil and criminal cases, in the first instance and on appeal, we follow the principle of party presentation. That is, we rely on the parties to frame the issues for decision and assign to courts the role of neutral arbiter of matters the parties present." Greenlaw v. United States, 554 U.S. 237, 243, 128 S.Ct. 2559, 171 L.Ed.2d 399 (2008). Although courts retain "the independent power to identify and apply the proper construction of governing law," Kamen v. Kemper Fin. Servs., Inc., 500 U.S. 90, 99, 111 S.Ct. 1711, 114 L.Ed.2d 152 (1991), "our adversary system is designed around the premise that the parties know what is best for them, and are responsible for advancing the facts and arguments entitling them to relief," Greenlaw, 554 U.S. at 244, 128 S.Ct. 2559 (quotation marks omitted). Accordingly, the Father has waived any argument with respect to the child's "acclimation." See, e.g., JP Morgan Chase Bank v. Altos Hornos de Mexico, S.A. de C.V., 412 F.3d 418, 428 (2d Cir.2005) (arguments not presented in an appellant's opening brief are waived).

CONCLUSION

To summarize:

Accordingly, because the District Court did not err regarding the first step of the Gitter test, and because the petitioner has not preserved any argument regarding the second step, the judgment dismissing the petition is AFFIRMED.