That's a lot of variables... what do I do?

Let \(\psi\) be defined as a function such that for any positive integer \(n\), \(\psi (n)\) finds the smallest possible integral value for \(m \geq 2\) such that \(n^{m} \pmod{10}=n \pmod{10}\). Let \(P\) be the probability that if a random nonnegative integer \(n\) is selected that \(\psi (n)=5\). If \(P\) can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime, positive integers, find \(a+b\).

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