\[\large L = \lim_{n \to \infty} \dfrac{1}{\sqrt[n]{n!}} \log_a \left( \displaystyle \sum_{k=1}^{a^n} (1+k)^{a^{-n}} \right) \]

Let \(a>1\) be any positive integer, find the value of \(\lfloor 100L \rfloor\).

\[\] **Notation**: \( \lfloor \cdot \rfloor \) denotes the floor function.

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