
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 $xOy$, with the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C_{1}$ is $\rho\sin \theta=2$. $M$ is a moving point on curve $C_{1}$, and point $P$ is on the line segment $OM$, satisfying $|OM||OP|=4$.
$(1)$ Find the Cartesian equation of the trajectory $C_{2}$ of point $P$;
$(2)$ The parametric equation of line $l$ is $\begin{cases} x=t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (where $t$ is the parameter), where $0 \leqslant \alpha < \pi$. Line $l$ intersects $C_{2}$ at point $A$, with $|OA|= \sqrt {3}$. Find the slope of line $l$.

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