
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 $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true.

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