
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 $xOy$, the parametric equation of the curve $C$ is $\begin{cases} x=4\cos a + 2 \\ y=4\sin a \end{cases} (a$ is the parameter$)$, and in the polar coordinate system with pole $O$ and the non-negative $x$-axis as the polar axis, the polar equation of the line $l$ is $\theta= \dfrac{\pi}{6} (\rho \in \mathbb{R})$.

$(1)$ Find the polar equation of the curve $C$;

$(2)$ Suppose the line $l$ intersects the curve $C$ at points $A$ and $B$, find the value of $|AB|$.

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