Maximum sum power of cubic

$\large{ \begin{cases} a+b+c+d+e+f=6 \\ a^2+b^2+c^2+d^2+e^2+f^2=\frac{36}5 \end{cases}}$

Let $a,b,c,d,e$ and $f$ be positive real numbers such that the system of equations above are fulfilled. If the maximum value of

$a^3+b^3+c^3+d^3+e^3 + f^3$

can be expressed as $\dfrac xy$ for coprime positive integers $x$ and $y$ , find the value of $\sqrt{x+y}$ .