A calculus problem by Hobart Pao

Calculus Level 3

Given \( z=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+\sqrt{x+\cdots }}}}} \), if \( \left.\dfrac{\partial z}{\partial x}\right|_{(x, y,z)=(2, 2,2)} \) can be represented in the form \( \dfrac{a}{b} \), where \(a, b\) are coprime positive integers, find \( a+b\).

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