# The sum is 1

Algebra Level 3

$6a^3+6b^3+6c^3+6d^3\geq a^2+b^2+c^2+d^2+N$ is always true for $$N$$, if $$a, b, c, d$$ are positive numbers, such that $$a+b+c+d=1$$.

If the maximum value of $$N$$ can be expressed in a $$\dfrac{x}{y}$$ formula, where $$\text{gcd}(x, y)=1$$, then find the value of $$x+y$$.

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