
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 an ordered pair of positive integers  $(m, n)$  be called *regimented* if for all nonnegative integers  $k$ , the numbers  $m^k$  and  $n^k$  have the same number of positive integer divisors. Let  $N$  be the smallest positive integer such that  $\left(2016^{2016}, N\right)$  is regimented. Compute the largest positive integer  $v$  such that  $2^v$  divides the difference  $2016^{2016}-N$ .

*Proposed by Ashwin Sah*

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