
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 \( 0 \leqslant x_{0} < 1 \), for all integers \( n > 0 \), let
$$
x_{n}=\left\{
\begin{array}{ll}
2 x_{n-1}, & \text{if } 2 x_{n-1} < 1 \\
2 x_{n-1}-1, & \text{if } 2 x_{n-1} \geqslant 1
\end{array}
\right.
$$
Determine the number of values of \( x_{0} \) such that \( x_{0} = x_{5} \).

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