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a2b3c4\large a^2b^3c^4a2b3c4
Positive real numbers aaa, bbb and ccc are such that a+b+c=18a+b+c=18a+b+c=18. If the maximum value of expression above can be written as p1α⋅p2βp_1^\alpha \cdot p_2^\beta p1α⋅p2β, where p1p_1p1 and p2p_2p2 are prime numbers and α\alphaα, β\betaβ are positive integers, find p1+p2+α+βp_1+p_2+\alpha+\betap1+p2+α+β.
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