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 $p_1+p_2+\alpha+\beta$ .