For classical inequalities lovers

Algebra Level 4

Let $a, b$ and $c$ positive real numbers such that $\dfrac{1}{a^2} +\dfrac{1}{b^2} +\dfrac{1}{c^2}=9$ . If $M$ is the maximum value of the following expression $\frac{1}{2a+b} +\frac{1}{2b+c} +\frac{1}{2c+a},$ enter $M^2$ as your answer.