Maximize it!

Algebra Level 4

\[\large a^2b^3c^4\]

Positive real numbers \(a\), \(b\) and \(c\) are such that \(a+b+c=18\). If the maximum value of expression above can be written as \(p_1^\alpha \cdot p_2^\beta \), where \(p_1\) and \(p_2\) are prime numbers and \(\alpha\), \(\beta\) are positive integers, find \(p_1+p_2+\alpha+\beta\).

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