
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 \( n = 2^b (2c+1) \). We call \( 2c+1 \) the odd part of \( n \). Given an odd integer \( n > 0 \), define the sequence \( a_0, a_1, a_2, \ldots \) as follows: \( a_0 = 2^n - 1 \), and \( a_{k+1} \) is the odd part of \( 3a_k + 1 \). Find \( a_n \).

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