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Algebra Level 5

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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