
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 \(0 \leq x_0 < 1\), let 
\[
x_n = \left\{ \begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1
\end{array} \right.
\]
for all integers \(n > 0\). Determine the number of initial values of \(x_0\) that satisfy \(x_0 = x_6\).

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