
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 ellipse $C$.
2. Find the equation of the ellipse with foci $F_1$ and $F_2$ whose major axis is the shortest possible, given it has a common point $M$ with the line $l$.

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