
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 Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. In this 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 line $l: \rho\sin(\theta - \frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$ with the polar axis. Find:  
(1) The rectangular coordinate equation of line $l$.  
(2) The polar coordinate equation of circle C.

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