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Calculus Level 4

You are given the curve \(\sin(x) + \sin(y) = 1\) lying in the first quadrant. Given that for some constant \(\alpha\), the limit \( \displaystyle \lim_{x\to0} x^\alpha \dfrac{d^2 y}{dx^2} \) exists and is non-zero. Find \(2\alpha\).

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