Sum of arctangents equals to pi/4, part 2

Geometry Level 5

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

Let \(a,b,c, d\) be positive integers where \(1 < a \le b\le c \le d \) such that the equation above is fulfilled. What is the largest possible value of \(d\)?

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