
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 $\begin{cases} x = 3\cos \theta \\ y = \sin \theta \end{cases}$ ($\theta$ as the parameter), and the parametric equation of line $l$ is $\begin{cases} x = a + 4t \\ y = 1 - t \end{cases}$ ($t$ as the parameter).
$(1)$ If $a=-1$, find the coordinates of the intersection points of $C$ and $l$;
$(2)$ If the maximum distance from a point on $C$ to $l$ is $\sqrt{17}$, find $a$.

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