
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 
\[x_n=\left\{
\begin{array}{ll} 
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1}-1 & \text{if } 2x_{n-1} \ge 1 
\end{array}
\right.\]
for all integers $n > 0$. Determine how many distinct values of $x_0$ are there such that $x_0 = x_6$?
A) 0
B) 31
C) 63
D) 64
E) 128

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