
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
Does the series \(\sum_{n=2}^\infty \frac{1}{\ln (n!)}\) converge? 

Does the series \(\frac{1}{3} + \frac{1}{3 \cdot 3^{1/2}} + \frac{1}{3 \cdot 3^{1/2} \cdot 3^{1/3}} + \ldots + \frac{1}{3 \cdot 3^{1/2} \cdot 3^{1/3} \ldots 3^{1/n}} + \ldots\) converge?

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