Define the function $f(x)=\frac{a}{1-x}$. If $f(-1)=f^{-1}(4a+1)$, find the product of all possible values of $a$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Define the function $f(x)=\frac{a}{1-x}$. If $f(-1)=f^{-1}(4a+1)$, find the product of all possible values of $a$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-\frac{1}{2}
Define the function $f(x)=\frac{a}{1-x}$. If $f(-1)=f^{-1}(4a+1)$, find the product of all possible values of $a$.
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