
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 
\[x_n=\left\{ \begin{array}{ll} 3x_{n-1} &\text{ if }3x_{n-1}<1 \\ 3x_{n-1}-1 &\text{ if }1\leq 3x_{n-1}<2 \\ 3x_{n-1}-2 &\text{ if }2\leq 3x_{n-1}<3 \end{array}\right.\]
for all integers $n>0$. For how many $x_0$ is it true that $x_0=x_6$?
- A) 243
- B) 486
- C) 729
- D) 1023

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