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If $x$ and $y$ are real values such that $0^{\circ} \leq x-y \leq 180^{\circ}$ and

$\begin{aligned} \sin x + \sin y &= 1, \\ \cos x + \cos y & = \sqrt{2}, \end{aligned}$

what is the value (in degrees) of $x - y$?

This problem is posed by Josh P.

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