
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, a circle $C$ passes through the point $P(\sqrt{2}, \frac{\pi}{4})$. The center of the circle is at the intersection of the polar axis and the line $\rho \sin (\theta - \frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$.
(1) Find the polar coordinate equation of circle $C$;
(2) Find the length of the chord intercepted by the line $\theta = \frac{\pi}{3} (\rho \in R)$ on circle $C$.

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