Algebra Level 4

Positive real numbers $a$, $b$, and $c$ are such that $\dfrac{1}{a}+ \dfrac{1}{b}+ \dfrac{1}{c} =a+b+c$. And if

$\frac{1}{(2a+b+c)^2}+ \frac{1}{(a+2b+c)^2}+ \frac{1}{(a+b+2c)^2} \leq \dfrac{p}{q}$

where $p$ and $q$ are coprime positive integers. Find $p+q$.

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