
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
The equation of line $l$ is $y = x + 2$. For any point $P$ on line $l$, if an ellipse is drawn such that its foci are the foci of the hyperbola $12x^2 - 4y^2 = 3$, then what is the equation of the ellipse with the shortest major axis?

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