# Hidden Inequalities

Geometry Level 3

In a triangle $$ABC$$, if $$\tan \frac{A}{2}$$, $$\tan \frac{B}{2}$$, and $$\tan \frac{C}{2}$$ are in a harmonic progression, then the minimum value of $$\cot \frac{B}{2}$$ can be expressed as $$\sqrt x$$, where $$x$$ is a square-free integer. Find $$x$$.

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