
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 compete in a race, each driving on separate elliptical tracks from time \( t = 0 \) to \( t = \infty \). Rational Man follows the path given by:
\[
x = 2\cos \left(t\right), \quad y = \sin \left(t\right)
\]
And Irrational Man follows the path given by:
\[
x = 1 + 3\cos \left(\frac{t}{2}\right), \quad y = 3\sin \left(\frac{t}{2}\right)
\]
Determine the smallest possible distance between a point \(A\) on Rational Man's track and a point \(B\) on Irrational Man's track.

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