Maximising Can Also Be Difficult!

Algebra

$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} = a+b+c$

Let $a,b,c$ be positive real numbers such that the above equation is satisfied. If the maximum value of the expression below is in the form of $\frac {m}{n} ,$ where $m,n$ are coprime positive integers, what is $m+n?$

$\dfrac{1}{\left(2a+b+c\right)^2}+\dfrac{1}{\left(2b+c+a\right)^2}+\dfrac{1}{\left(2c+b+a\right)^2}$