Consider integers \(a\) and \(P\) that satisfy the equations
###### Image Credit: Wikimedia Pythagoras Theorem Proof

\[ \begin{cases} (4a+5)^2 = a^2 + [4(a+1)]^2 \\ P =(271a)^2 + [1084(a+1)]^2\\ \end{cases} \]

Find sum of all value(s) of \(P\).

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