Sum of arctangents equals to pi/4

Geometry Level 5

\[\arctan\left(\frac{1}{a}\right)+\arctan\left(\frac{1}{b}\right)+\arctan\left(\frac{1}{c}\right)=\frac{\pi}{4}\]

Let \(a,b,c\) be positive integers where \(1 < a \le b\le c \) such that the equation above is fulfilled. If we denote \(P = a\times b\times c\), find the sum of all possible values of \(P\).

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