
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
Given the line $l: x-y+9=0$ and the ellipse $C: \begin{cases} x=2\sqrt{3}\cos\theta \\ y=\sqrt{3}\sin\theta \end{cases}$ (with $\theta$ as the parameter).

$(1)$ Find the coordinates of the two foci $F_1$ and $F_2$ of the ellipse $C$;  
$(2)$ Find the equation of the ellipse with the shortest major axis that has $F_1$ and $F_2$ as its foci and shares a common point $M$ with the line $l$.

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