A *binary string* is a string whose characters are only `0`

s and/or `1`

s. For example, \(0101, 11111, 110001\) are binary strings.

A string \(s\) is *periodic* if there exists another string \(t\) whose length is less than \(s\) such that concatenating several copies of \(t\) produces \(s\). For example, \(0101\) is periodic (\(t = 01\)), and \(1111\) is periodic (\(t = 11\) or \(t = 1\)), but \(10101\) is not periodic. A string is *aperiodic* if it is not periodic.

This problem asks about the number of aperiodic binary strings of length \(23\). For this problem, compute the number of aperiodic binary strings of length \((23!)^{23}\) (that is, 23 factorial raised to the 23th power). Enter your answer modulo \(10^9 + 7 = 1000000007\).

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