
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
An eccentric mathematician has a ladder with \( n \) rungs that he always ascends and descends in the following way: When he ascends, each step he takes covers \( a \) rungs of the ladder, and when he descends, each step he takes covers \( b \) rungs of the ladder, where \( a \) and \( b \) are fixed positive integers. By a sequence of ascending and descending steps he can climb from ground level to the top rung of the ladder and come back down to ground level again. Find, with proof, the minimum value of \( n \), expressed in terms of \( a \) and \( b \).

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