Triangle + Trigonometry = Anglometry

Geometry Level 3

In any triangle \(ABC\), if \( \cot A = \sqrt{ac} , \cot B = \sqrt{c/a}, \cot C = \sqrt{ a^2 / c } \), then the value of \( a + a^2 + c \) is equal to \( \underline{ \quad } \)?

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