
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let   $f: \mathbb{R}\to\mathbb{R}$  is a function such that  $f( \cot x ) = \cos 2x+\sin 2x$  for all  $0 < x < \pi$ . Define  $g(x) = f(x) f(1-x)$  for  $-1 \leq x \leq 1$ . Find the maximum and minimum values of  $g$  on the closed interval  $[-1, 1].$ 

Present the answer in LaTex format: \boxed{Your answer}
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