Consider all polynomials \(f(x)\) with integer coefficients and degree at most 100. There are \(N_f\) distinct integer values for which \(f(n) = 2\), and \(M_f\) distinct integer values for which \(f(m)=-2\).

Over all such polynomials, what is the maximum possible value of \(N_f \times M_f\)?

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