
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
For two positive integers \( a \) and \( b \), Ivica and Marica play the following game: Given two piles of \( a \) and \( b \) cookies, on each turn a player takes \( 2n \) cookies from one of the piles, of which they eat \( n \) and put \( n \) of them on the other pile. Number \( n \) is arbitrary in every move. Players take turns alternatively, with Ivica going first. The player who cannot make a move loses. Assuming both players play perfectly, determine all pairs of numbers \( (a, b) \) for which Marica has a winning strategy.

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