
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
For positive integers  $n$  and  $k$ , let  $\mho(n,k)$  be the number of distinct prime divisors of  $n$  that are at least  $k$ . For example,  $\mho(90, 3)=2$ , since the only prime factors of  $90$  that are at least  $3$  are  $3$  and  $5$ . Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

*Proposed by Daniel Zhu.*

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