
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 curve $C_1$ is $\begin{cases} x=1+2\cos \alpha \\ y= \sqrt {3}\sin \alpha \end{cases}$ ($\alpha$ is the parameter). By reducing all the x-coordinates of points on curve $C_1$ to half of their original values and all the y-coordinates to $\frac{\sqrt{3}}{3}$ of their original values, we obtain curve $C_2$. In the polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $4\rho\sin (\theta+ \frac{\pi}{3})+ \sqrt{3}=0$.
1. Find the polar equation of curve $C_2$ and the polar coordinates of the intersection points between line $l$ and curve $C_2$;
2. Let point $P$ be any point on curve $C_1$. Find the maximum distance from point $P$ to line $l$.

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