Let $a,$ $b,$ $c$ be positive real numbers such that
\[\log_a b + \log_b c + \log_c a = 0.\]Find
\[(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ $c$ be positive real numbers such that
\[\log_a b + \log_b c + \log_c a = 0.\]Find
\[(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
3
Let $a,$ $b,$ $c$ be positive real numbers such that
\[\log_a b + \log_b c + \log_c a = 0.\]Find
\[(\log_a b)^3 + (\log_b c)^3 + (\log_c a)^3.\]
train