
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
Given \(0 \leq x_{0} < 1\), for all integers \(n > 0\), let 
\[ x_{n} = \begin{cases} 
2x_{n-1}, & \text{if } 2x_{n-1} < 1, \\ 
2x_{n-1} - 1, & \text{if } 2x_{n-1} \geq 1.
\end{cases} \]
Determine the number of values of \( x_{0} \) for which \( x_{0} = x_{5} \) holds.

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