
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. It is known that curve $C$ is defined by $\rho=4\cos\theta$, and line $l$ passes through point $M(1,0)$ with an inclination angle $\alpha= \frac{\pi}{6}$.  
(1) Convert the polar equation of curve $C$ into a Cartesian coordinate equation, and write the parametric equation of line $l$;  
(2) If line $l$ 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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