
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 
\[ x_n = \left\{ 
\begin{array}{ll} 
3x_{n-1}     & \text{if } 3x_{n-1} < 1 \\ 
3x_{n-1} - 2 & \text{if } 3x_{n-1} \ge 1 
\end{array} 
\right. \]
for all integers \( n > 0 \). For how many \( x_0 \) is it true that \( x_0 = x_3 \)?
A) 9
B) 18
C) 27
D) 36
E) 45

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