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Calculus Level 4

\[\large \large \lim_{x\to0}\dfrac{\ln(1+\sin^3x \cos^2x)\cot(\ln^3(1+x))\tan^4x}{\sin(\sqrt{x^2+2}-\sqrt{2})\ln(1+x^2)} \]

If the limit above is equal to \( a\sqrt b \), where \(a\) and \(b\) are integers with \(b\) square-free, find \(a+b\).

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