
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, it is known that circle $C$ passes through point $P(\sqrt{2}, \frac{\pi}{4})$, and the center of the circle is the intersection point of the line $\rho\sin(\theta - \frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$ with the polar axis. Then, the polar equation of circle $C$ is \_\_\_\_\_\_.

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