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Algebra Level 4

Given \( 100\) reals \( a_{1}, a_{2},\ldots , a_{100} \) such that \(\displaystyle \sum ^{100}_{i=1}a^{16}_{i} = 2^{32}\), we have maximum value of \(\displaystyle \sum ^{100}_{i=1}a^{17}_{i}\) to be of the form \( a^{k} \), where \( a \) is a prime and \( k \) is a positive integer. Find \( (a+k) \).

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