
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 plane rectangular coordinate system $xOy$, $C_{1}$: $\left\{{\begin{array}{l}{x=\frac{{1-{t^2}}}{{1+{t^2}}}}\\{y=\frac{{{{(1+t)}^2}}}{{1+{t^2}}}}\end{array}}\right.$ (where $t$ is a parameter), with the coordinate origin as the pole and the positive half-axis of the $x$-axis as the polar axis, establishes a polar coordinate system. The curve $C_{2}$: $ρ=2\sqrt{3}\cosθ$, and the ray $l$: $θ=\frac{π}{6}$ ($ρ>0$).<br/>$(1)$ Find the range of the horizontal and vertical coordinates of $C_{1}$, and convert $C_{1}$ into polar coordinates;<br/>$(2)$ If the intersection point of $C_{1}$ and the $y$-axis is $P (not the origin)$, and the ray $l$ intersects $C_{1}$ and $C_{2}$ at points $A$ and $B$ respectively, find the area of $\triangle PAB$.

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