
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 
\[
x_n=\begin{cases} 
2x_{n-1} + 0.1 & \text{if } 2x_{n-1}<1 \\ 
2x_{n-1}-1 + 0.1 & \text{if } 2x_{n-1}\ge 1 
\end{cases}
\]
for all integers $n>0$. Determine for how many $x_0$ it is true that $x_0=x_7$.
A) 0
B) 31
C) 63
D) 127
E) 255

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