
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 the curve $C$ is 
$$
\begin{cases}
x=2\cos \theta \\
y=4\sin \theta 
\end{cases}
(\theta \text{ is the parameter}),
$$
and the parametric equation of the line $l$ is 
$$
\begin{cases}
x=1+t\cos \alpha \\
y=2+t\sin \alpha 
\end{cases}
(t \text{ is the parameter}).
$$
$(1)$ Find the rectangular coordinate equations of $C$ and $l$;
$(2)$ If the midpoint of the line segment obtained by intersecting the curve $C$ with line $l$ is $(1,2)$, find the slope of $l$.

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