Given that the equation $\displaystyle ax^3-x^2+ax-b=0$ ($\displaystyle a\neq 0,b\neq 0$) has three positive real roots. Determine the greatest value of the following expression

$\displaystyle P=\frac{11a^2-3\sqrt3ab-\frac{1}{3}}{9b-10\left ( \sqrt3a-1 \right )}$