
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 $xOy$ plane of the Cartesian coordinate system, the parametric equations of the line $l$ are $\left\{\begin{array}{l}{x=\frac{1}{2}t}\\{y=\frac{\sqrt{3}}{2}t}\end{array}\right.$ (where $t$ is the parameter). Taking the coordinate origin as the pole and the non-negative $x$-axis as the polar axis, the polar coordinate equation of curve $C$ is $\rho ^{2}-2\rho \cos \theta -2=0$. The polar coordinates of point $P$ are ($\frac{2\sqrt{15}}{3}$, $\frac{2\pi}{3}$).<br/>$(1)$ Find the polar coordinate equation of line $l$ and the distance from point $P$ to line $l$;<br/>$(2)$ If line $l$ intersects curve $C$ at points $M$ and $N$, find the area of $\triangle PMN$.

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