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Algebra Difficulty 4.0 AIME Find the answer United Kingdom

Find all triples $(x, y, z)$ of real numbers satisfying the equations
$x^2 + 2yz = 4, \qquad y^2 + 2zx = 4, \qquad z^2 + 2xy = 1.$

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; question papers published free at bmos.ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.