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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