Hotter and Fresher

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

This problem is extracted from the 2016 Vietnamese University Entrance Examination which just took place a few hours ago. Solutions and discussions are always welcome!