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Algebra Difficulty 4.0 AIME Find the answer United Kingdom

Find all real values of $x$ , $y$ and $z$ such that
$(x + 1)yz = 12, \quad (y + 1)zx = 4 \quad \text{and} \quad (z + 1)xy = 4.$

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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.