
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
Given the polar equation of curve C is $\rho - 6\cos\theta + 2\sin\theta + \frac{1}{\rho} = 0$, with the pole as the origin of the Cartesian coordinate system and the polar axis as the positive x-axis, establish a Cartesian coordinate system in the plane xOy. The line $l$ passes through point P(3, 3) with an inclination angle $\alpha = \frac{\pi}{3}$.  
(1) Write the Cartesian equation of curve C and the parametric equation of line $l$;  
(2) Suppose $l$ intersects curve C at points A and B, find the value of $|AB|$.

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