
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
Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time \( t = 0 \) to \( t = \infty \). Rational Man drives along the path parameterized by
\[
x = \cos t, \quad y = \sin t,
\]
and Irrational Man drives along the path parameterized by
\[
x = \cos \left(\frac{t}{2}\right), \quad y = \sin \left(\frac{t}{2}\right).
\]
If \( A \) is a point on Rational Man's racetrack, and \( B \) is a point on Irrational Man's racetrack, then find the smallest possible distance \( AB \).

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