
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 both buy new bikes, and they decide to drive around two racetracks from time $t = 0$ to $t = \infty.$ Rational Woman drives along the path parameterized by
\begin{align*}
x &= 2 + 2 \cos t, \\
y &= 1 + 2 \sin t,
\end{align*}
and Irrational Woman drives along the path parameterized by
\begin{align*}
x &= 3 + 3 \cos \frac{t}{2}, \\
y &= 2 + 3 \sin \frac{t}{2}.
\end{align*}
If $A$ is a point on Rational Woman's racetrack, and $B$ is a point on Irrational Woman's racetrack, then find the smallest possible distance $AB$.

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