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\[ \large f(x) = \sum_{n=0}^{\infty}\dfrac{2^{2n-1}(-1)^{n}x^{n}\ln^{2n}(x^{2}+1)\sin^{n}(x)}{(2n)!} \] For \(f(x)\) as defined above, find \[ \large \left \lfloor 10^{7} f\left(\dfrac{\pi}{2}\right)\right \rfloor. \]

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