
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
Find all functions  $f:\mathbb{R}\to\mathbb{R}$  such that
                          $f(f(x)f(1-x))=f(x)$     and      $f(f(x))=1-f(x)$ ,
for all real  $x$ .

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