Find the largest Integer, without programming!

Algebra Level 5

\[ \frac1{\sqrt{2+\sqrt3}} + \frac1{\sqrt{4+\sqrt{15}}} + \ldots + \frac1{\sqrt{2n+\sqrt{4n^2-1}}} < 2013 \sqrt2 \]

Find the largest positive integer \(n\) such that it satisfies the inequality above.

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