
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 \(0 \le x_0 < 1\), let 
\[
x_n = \begin{cases} 
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \geq 1 
\end{cases}
\]
for all integers \(n > 0\). For how many \(x_0\) is it true that \(x_0 = x_3\)?
A) 4
B) 8
C) 16
D) 32
E) 64

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