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

\[\begin{align} \sin x + \sin y &= 1, \\ \cos x + \cos y & = \sqrt{2}, \end{align} \]

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

This problem is posed by Josh P.

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