If $f(a) = \frac{1}{1-a}$, find the product $f^{-1}(a) \times a \times f(a)$.  (Assume $a \neq 0$ and $a \neq 1$.)
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
If $f(a) = \frac{1}{1-a}$, find the product $f^{-1}(a) \times a \times f(a)$.  (Assume $a \neq 0$ and $a \neq 1$.)
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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If $f(a) = \frac{1}{1-a}$, find the product $f^{-1}(a) \times a \times f(a)$.  (Assume $a \neq 0$ and $a \neq 1$.)
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