HMMT 2007 U19

Define $x \star y=\frac{\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$ . Compute

$((\cdots((2007 \star 2006) \star 2005) \star \cdots) \star 1) .$

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{\sqrt{15}}{9}

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