
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
There are 2009 piles, each containing 2 stones. You are allowed to take the largest pile with an even number of stones (if there are multiple such piles, you can choose any one of them) and transfer exactly half of its stones to any other pile. What is the maximum number of stones that can be obtained in a single pile through these operations?

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