# Hotter and Fresher

**Algebra**Level 4

\[\large P=3^{x+y-4}+(x+y+1)\cdot 2^{7-x-y}-3(x^2+y^2)\] 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 \(a+b\).