solution: We have that
  \[f(f(g(f(g(f(-x)))))) = f(f(g(f(g(-f(x)))))) = f(f(g(f(g(f(x)))))),\]so the function is $\boxed{\text{even}}.$
  
  More generally, if we have a composition of functions, and at least one of the functions is even, then the whole composition of functions is even.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $f(x)$ be an odd function, and let $g(x)$ be an even function.  Is $f(f(g(f(g(f(x))))))$ even, odd, or neither?
  
  Enter "odd", "even", or "neither".
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \text{even}
extra_info.question: Let $f(x)$ be an odd function, and let $g(x)$ be an even function.  Is $f(f(g(f(g(f(x))))))$ even, odd, or neither?
  
  Enter "odd", "even", or "neither".
extra_info.split: test