Maximizing the w

Algebra Level 5

Suppose $$w,x,y,z$$ satisfy \begin{align*}w+x+y+z&=25,\\wx+wy+wz+xy+xz+yz&=2y+2z+193\end{align*} The largest possible value of $$w$$ can be expressed in lowest terms as $$w_1/w_2$$ for some integers $$w_1,w_2>0$$. Find $$w_1+w_2$$.

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