
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
Let \( n \geq 3 \) be a fixed integer. The number 1 is written \( n \) times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers \( a \) and \( b \), replacing them with the numbers 1 and \( a + b \), then adding one stone to the first bucket and \( \operatorname{gcd}(a, b) \) stones to the second bucket. After some finite number of moves, there are \( s \) stones in the first bucket and \( t \) stones in the second bucket, where \( s \) and \( t \) are positive integers. Find all possible values of the ratio \(\frac{t}{s}\).

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