
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 $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=1+\frac{1}{2}t\\ y=\frac{\sqrt{3}}{2}t\end{array}\right.$ (where $t$ is a parameter). Taking $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $ρsin^{2}\frac{θ}{2}=1$.  
$(1)$ Find the rectangular coordinate equation of $C$ and the polar coordinates of the intersection points of $C$ with the $y$-axis.  
$(2)$ If the line $l$ intersects $C$ at points $A$ and $B$, and intersects the $x$-axis at point $P$, find the value of $\frac{1}{|PA|}+\frac{1}{|PB|}$.

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