
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 three piles of coins, each containing \(a\), \(b\), and \(c\) (\(a, b, c \geqslant 2015\)) coins respectively. Each time, the following operations can be chosen:

1. Choose a pile with an even number of coins, evenly divide its coins, and add the parts to the other two piles.
2. Choose a pile with at least 2017 coins and an odd number of coins, remove 2017 coins from it, and add 1009 coins to each of the other two piles.

Assuming there are enough backup coins, find all triplets \((a, b, c)\) such that after a finite number of operations, one of the piles will have at least \(2017^{2017}\) coins.

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