Consider the system \(x + y = z + u, 2xy = zu\). Find the greatest value of the real constant \(m\) such that \(m ≤ \frac{x}{y}\) for every positive integer solution \(x, y, z, u\) of the system where \(x ≥ y\).

Express your answer up to three digit places. This problem is not original.

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