
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 $C$ with center at point $C(2, \frac{\pi}{2})$ and radius $3$ intersects line $l$: $\theta= \frac{\pi}{3} (\rho=R)$ at points $A$ and $B$.  
$(1)$ Find the standard equations of circle $C$ and line $l$.  
$(2)$ Find the length of chord $|AB|$.

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