Maximize it!

Algebra Level 4

a2b3c4\large a^2b^3c^4

Positive real numbers aa, bb and cc are such that a+b+c=18a+b+c=18. If the maximum value of expression above can be written as p1αp2βp_1^\alpha \cdot p_2^\beta , where p1p_1 and p2p_2 are prime numbers and α\alpha, β\beta are positive integers, find p1+p2+α+βp_1+p_2+\alpha+\beta.

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