Arccot and Arctans!

Geometry Level 4

\[\displaystyle\large \cot^{-1}\left[\dfrac{ab+1}{a-b}\right]+\cot^{-1} \left[\dfrac{bc+1}{b-c}\right]+\cot^{-1}\left[\dfrac{ca+1}{c-a}\right] \]

If \(a,~b,~c\) are distinct non zero numbers having same sign, then the expression above returns two distinct values, \(m\) and \(n\), find the value of \(m+n\).

Note: This problem uses the convention \(0 < \cot^{-1}(x) < \pi\) for the range of the inverse cotangent function.

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