
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
Consider the sequences \( \left(a_{n}\right) \) and \( \left(b_{n}\right) \) defined by
\[
a_{1}=3, \quad b_{1}=100, \quad a_{n+1}=3^{a_{n}}, \quad b_{n+1}=100^{b_{n}}.
\]
Find the smallest integer \( m \) for which \( b_{m} > a_{100} \).

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