
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
In the polar coordinate system, where $O$ is the pole, it is known that the circle $C$ has its center at $(2, \frac{\pi}{3})$ and radius $r=1$. Point $P$ moves on circle $C$.  
$(1)$ Find the polar equation of circle $C$;  
$(2)$ In the Cartesian coordinate system (using the same unit of length as the polar coordinate system, and taking the pole $O$ as the origin, with the polar axis as the positive half-axis of $x$), if $Q$ is the midpoint of the line segment $OP$, find the Cartesian equation of the trajectory of point $Q$.

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