
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 rectangular coordinate system $xOy$, the parametric equation of the curve $C_1$ is $$\begin{cases} x=3\cos t \\ y=\sin t \end{cases}$$ (where $t$ is the parameter). Establish a polar coordinate system with the origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar equation of line $l$ is $\theta = \frac{\pi}{6}$ ($\rho \in \mathbb{R}$).

1. Find the polar equation of $C_1$.
2. If the polar equation of curve $C_2$ is $\rho + 8\cos\theta = 0$, and the intersection point of line $l$ and $C_1$ in the first quadrant is $A$, and the intersection point of $C_2$ and $l$ (excluding the origin) is $B$, find $|AB|$.

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