
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
A and B play the following game with N counters. A divides the counters into 2 piles, each with at least 2 counters. Then B divides each pile into 2 piles, each with at least one counter. B then takes 2 piles according to a rule which both of them know, and A takes the remaining 2 piles. Both A and B make their choices in order to end up with as many counters as possible.

There are 3 possibilities for the rule:
- **R1**: B takes the biggest heap (or one of them if there is more than one) and the smallest heap (or one of them if there is more than one).
- **R2**: B takes the two middling heaps (the two heaps that A would take under R1).
- **R3**: B has the choice of taking either the biggest and smallest, or the two middling heaps.

For each rule, how many counters will A get if both players play optimally?

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