
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
Let \(a_n\) be the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{k}\), where \(k = a_n\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Which values \(b_n\) occur for infinitely many \(n\)?

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