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Calculus Level 5

Let \(f\) be a twice differentiable function defined on \(x>0 \) satisfying \( x f(y) + y f(x) = f(xy) \) for all \(x,y> 0\). Given that \(f'(1) = 1\), find \(\displaystyle 4\sum_{x=1}^\infty f''(x) f''(x+2) \).

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