Problem

Problem

If the sum of all values of

x

satisfying the system of equations

\tan x + \tan y + \tan x \cdot \tan y = 5

\sin(x + y) = 4 \cos x \cdot \cos y

is

\frac{k\pi}{2}

, where

x \in \left(0, \frac{\pi}{2}\right)

then find the values of

k

.

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