
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 the sequence $\{a\_n\}$ that satisfies $a_{n+1}=\begin{cases} 2a\_n, & 0 \leqslant a\_n < \frac{1}{2} \\ 2a\_n - 1, & \frac{1}{2} \leqslant a\_n < 1 \end{cases}$, and if $a\_1 = \frac{6}{7}$, find the value of $a_{20}$.

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