
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  $S$  be the value of
\[\sum_{n=1}^\infty \frac{d(n) + \sum_{m=1}^{\nu_2(n)}(m-3)d\left(\frac{n}{2^m}\right)}{n},\]
where  $d(n)$  is the number of divisors of  $n$  and  $\nu_2(n)$  is the exponent of  $2$  in the prime factorization of  $n$ . If  $S$  can be expressed as  $(\ln m)^n$  for positive integers  $m$  and  $n$ , find  $1000n + m$ .

*Proposed by Robin Park*

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