
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 \( f: \mathbb{N} \rightarrow \mathbb{N} \) be a strictly increasing function such that \( f(1)=1 \) and \( f(2n) f(2n+1) = 9 f(n)^{2} + 3 f(n) \) for all \( n \in \mathbb{N} \). Compute \( f(137) \).

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