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Algebra Level 5

Real numbers \( a_1, a_2, \ldots, a_{80090} \) satisfy

\[ (a_1 + a_2 + \ldots + a_{80090} )^2 + a_1 ^2 + a_2^2 + \ldots + a_{80090}^2 \leq 80090. \]

What is the smallest integer value of \(N\) such that for all possible such tuples \((a_1,...,a_{80090})\), we have \( |a_i| \leq N \) for all \(i\)?

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