Given the equation $\begin{aligned}\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}\end{aligned}$, if the sum of all possible values of $x$ is $\frac{b}{a}$, find the value of $a+b$.

Let's think step by step and output the final answer within \boxed{}.
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