62208 and M

Algebra Level 3

\[62208\big(a^{7}+b^{7}+c^{7}+d^{7}\big)^{2}\le M\big(a^{2}+b^{2}+c^{2}+d^{2}\big)^{7}\]

What is the smallest positive integer \(M\) such that this inequality holds true for all \(a,b,c,d\) satisfying \(a,b,c,d\in \mathbb{R}\) and \(a+b+c+d=0?\)

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