(100 root 3) days streak problem

Calculus Level 5

\[\large \displaystyle \lim_{n\to\infty} \frac1{n^4} \left[ (n^2+1^2)(n^2+2^2)(n^2+3^2)\ldots (n^2+4n^2) \right]^{\frac 1n} \]

If the limit above equals to \( ae^{b \tan^{-1}(c) - bc} \) for integers \(a,b,c\) with \(\displaystyle e =\sum_{m=0}^\infty \frac1{m!} \), calculate \(a+b+c\).

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