Let $a$ and $b$ be nonzero real constants such that $|a| \neq |b|.$  Find the number of distinct values of $x$ satisfying
\[\frac{x - a}{b} + \frac{x - b}{a} = \frac{b}{x - a} + \frac{a}{x - b}.\]
<|im_start|>system
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<|im_start|>user
Let $a$ and $b$ be nonzero real constants such that $|a| \neq |b|.$  Find the number of distinct values of $x$ satisfying
\[\frac{x - a}{b} + \frac{x - b}{a} = \frac{b}{x - a} + \frac{a}{x - b}.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
3
Let $a$ and $b$ be nonzero real constants such that $|a| \neq |b|.$  Find the number of distinct values of $x$ satisfying
\[\frac{x - a}{b} + \frac{x - b}{a} = \frac{b}{x - a} + \frac{a}{x - b}.\]
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