
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
Suppose there are three empty buckets and  $n \ge 3$  marbles. Ani and Budi play a game. For the first turn, Ani distributes all the marbles into the buckets so that each bucket has at least one marble. Then Budi and Ani alternate turns, starting with Budi. On a turn, the current player may choose a bucket and take 1, 2, or 3 marbles from it. The player that takes the last marble wins. Find all  $n$  such that Ani has a winning strategy, including what Ani's first move (distributing the marbles) should be for these  $n$ .

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