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Given that

$\large S = \int_{-1}^{1}\frac{x^{4}(1-x)^{4}}{1+x^{2}} \, dx, \quad \quad M = \int_{0}^{1}\frac{x^{6}}{1+x^{2}}\, dx$

Find the value of $\lfloor S-12M \rfloor$.

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.

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