Limit It!

Calculus Level 5

Let \(x_1,x_2, \mbox{ and } x_3\) be roots of the equation \((x-1)(x-8)(x-31)=1\)

If \[\lim\limits_{m\rightarrow\infty}\lim\limits_{n\rightarrow\infty}\sum\limits_{i=1}^{3}\cos^m\left(\pi n! x_i\right)=k+7\]

for integers \(m\) and \(n\), find the value of \(k\).

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