Let \(\sigma\) be a permutation of \(\{1, 2, 3, \ldots, 100\}\). Then there exists a smallest positive integer \(f(\sigma)\) such that \(\sigma^{f(\sigma)} = \text{id}\), where \(\text{id}\) is the identity permutation.

What is the largest value of \(f(\sigma)\) over all \(\sigma?\)

\(\)

**Note:** \(\sigma^k\) is \(\sigma\) applied \(k\) times in succession.

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