Let
\[f(x) = \frac{2x + 3}{kx - 2}.\]Find all real numbers $k$ so that $f^{-1}(x) = f(x).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let
\[f(x) = \frac{2x + 3}{kx - 2}.\]Find all real numbers $k$ so that $f^{-1}(x) = f(x).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
(-\infty,-\frac{4}{3}) \cup (-\frac{4}{3},\infty)
Let
\[f(x) = \frac{2x + 3}{kx - 2}.\]Find all real numbers $k$ so that $f^{-1}(x) = f(x).$
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