
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
\[
\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}
\]

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