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Let \(a\) be a positive integer divisible by \(4\) but not divisible by \(8\). What is the largest positive integer \(n\) such that \(2^n\) divides

\[a^{100} + a^{101} + a^{102} + a^{103} + \cdots +a^{1000}?\]

This problem is posed by Kiriti M.

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