# Hotter and Fresher

Algebra Level 5

$\large P=3^{x+y-4}+(x+y+1)\cdot 2^{7-x-y}-3(x^2+y^2)$ Given that $$x$$ and $$y$$ are real numbers satisfying $x+y+1=2(\sqrt{x-2}+\sqrt{y+3}).$

If the maximum value of $$P$$ can be expressed as $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, find $$a+b$$.

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