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

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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