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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