
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
Any positive integer \( n \) can be written in the form \( n=2^{a} q \), where \( a \geq 0 \) and \( q \) is odd. We call \( q \) the odd part of \( n \). Define the sequence \( a_0, a_1, \ldots \), as follows: \( a_0 = 2^{2011} - 1 \) and for \( m \geq 0 \), \( a_{m+1} \) is the odd part of \( 3a_m + 1 \). Find \( a_{2011} \).

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