
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

The distance between points A and B is 36 kilometers. Two superheroes, Alpha and Beta, begin walking simultaneously from point A to point B. Upon reaching point B, they immediately start walking back to point A, and upon reaching point A, they immediately walk back to point B, continuously moving back and forth between A and B. If Alpha walks at a speed of \(2k\) kilometers per hour and Beta walks at a speed of \(k\) kilometers per hour. Let \(p\) be the number of hours after which the distance between Alpha and Beta reaches its maximum for the 2012th time, and \(q\) be the number of hours after which the distance between Alpha and Beta reaches its maximum for the 2013th time. If \(q-p\) is a positive integer, find the maximum value of the positive integer \(k\).

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