
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
A little rascal ran a race with a horse mounted on a mechanical carousel. After \(a\) seconds, she found that the horse, having made a circle, had caught up to her. Instantly turning around, the little rascal ran towards the horse at the same speed and met it after \(\frac{a}{2}\) seconds. Determine how long it takes for the carousel to make a full rotation, assuming all movements are uniform.

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