
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 Woman and Irrational Woman decide to drive around two racetracks. Rational Woman drives on a path parameterized by
\[
x = 2 + \cos t, \quad y = \sin t,
\]
and Irrational Woman drives on a path parameterized by
\[
x = 3 + 3 \cos \frac{t}{2}, \quad y = \sin \frac{t}{2}.
\]
Find the smallest possible distance between any two points, each belonging to one of the tracks.

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