
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 $\{a_n\}$ satisfying
$$
a_{n+1} = 
\begin{cases} 
2a_{n} & (0 \leq a_n < \frac{1}{2}) \\
2a_{n}-1 & (\frac{1}{2} \leq a_n < 1)
\end{cases},
$$
if $a_1 = \frac{6}{7}$, calculate the value of $a_{2011}$.

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