
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 $C$ is $\begin{cases}x=2\cos \theta \\ y=4\sin \theta\end{cases} (\theta$ is the parameter$)$, and the parametric equation of line $l$ is $\begin{cases}x=1+t\cos \alpha \\ y=2+t\sin \alpha\end{cases} (t$ is the parameter$)$.

$(1)$ Find the Cartesian equations of $C$ and $l$;

$(2)$ If the midpoint of the segment cut by curve $C$ on line $l$ has coordinates $(1,2)$, find the slope of $l$.

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