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Algebra Difficulty 3.0 AMC 10/12 Prove it Canada

Suppose that, for some angles $x$ and $y$ , $\sin^2 x + \cos^2 y = \frac{3}{2}a$ and $\cos^2 x + \sin^2 y = \frac{1}{2}a^2$ . Determine the possible value(s) of $a$ .

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.