
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 $0 \leq x_0 < 1$, consider the sequence defined by:
\[x_n = \left\{
\begin{array}{ll}
2x_{n-1} & \text{if } 2x_{n-1} < 1 \\
2x_{n-1} - 1 & \text{if } 2x_{n-1} \ge 1 \\
\end{array}\right.\]
for all integers $n > 0$. Determine how many $x_0$ satisfy $x_0 = x_{10}$.
A) 510
B) 511
C) 1023
D) 1024

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