
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
Define a function \( f \) on the real numbers by

\[ 
f(x) = \begin{cases} 
2x & \text{if } x < \frac{1}{2} \\
2x - 1 & \text{if } x \geq \frac{1}{2} 
\end{cases}
\]

Determine all values \( x \) satisfying \( f(f(f(f(f(x))))) = x \).

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