Algebra Level 4

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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