
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 a positive integer  $n$ , define  $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$  and let  $g:\mathbb N\rightarrow \mathbb Q$  be a function such that  $\sum_{d\mid n}g(d)=f(n)$  for all positive integers  $n$ . Given that  $g(12321)=\frac{p}{q}$  for relatively prime integers  $p$  and  $q$ , find  $v_2(p)$ .

*Proposed by Michael Ren*

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