AM-GM Won't Work Here

Algebra Level 5

$\large x^5 y + y^5 z + z^5 x$

Let $x, y,$ and $z$ be non-negative reals such that $x+y+z=1$ .

The maximum value of the above expression can be represented as $\dfrac {a^b}{c^d}$ , where $a$ and $c$ are not perfect powers, and $a,b,c,d$ are positive integers. Find the value of $a+b+c+d$ .

Bonus: Generalize this for the expression $x^n y + y^n z + z^n x$ .

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