Let \(x,y,z\) be positive real numbers satisfying \(x,y,z > 1\). It is true that \[\log_y(xz)\cdot\log_z(xy) \ge k\] for some real number \(k\). Find \(k\) accurate to 3 decimal places.

**Details and Assumptions**

\(k\) may possibly be an integer.

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