(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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