
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 are biking on two tracks from $t = 0$ to $t = \infty$. Rational Woman bikes along the path parameterized by:
\[
x = 2 \cos t, \quad y = 2 \sin t
\]
Irrational Woman bikes along the path parameterized by:
\[
x = 3 + 3 \cos \frac{t}{2}, \quad y = 3 \sin \frac{t}{2}
\]
Find the smallest possible distance between a point $C$ on Rational Woman's track and a point $D$ on Irrational Woman's track.

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