\[ \large \displaystyle \int_0^{10} \left \lfloor \left ( \frac{4|x|}{x^2+16} \right)^m \right \rfloor \, dx \]

Given that 1 lies between the roots (of \(y\)) of the equation \(y^2-my + 1 = 0 \), evaluate the integral above.

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