HMMT 2019 A7

Find the value of

$\sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \sum_{c=1}^{\infty} \frac{a b(3 a+c)}{4^{a+b+c}(a+b)(b+c)(c+a)}$

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{1}{54}

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