100 Degree Function

Suppose f(x)f(x) is a polynomial with integer coefficients of degree 100100. Find the biggest possible number of pairs of integers n<mn<m, such that f(n)=mf(n)=m and f(m)=nf(m)=n.

Details and assumptions

You are asked to find the biggest possible number of pairs, not the biggest pair. Hence, your answer is just an integer, not a pair of integers.

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