
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 that the center of an ellipse is at the origin of the coordinate, the focus is on the x-axis, the focal length is $2\sqrt{3}$, and the length of the major axis is twice that of the minor axis. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively.

(I) Find the standard equation of the ellipse;

(II) If the line $l$ passing through the upper vertex of the ellipse and the left focus intersects the ellipse at points $P$ and $Q$, find the area of the triangle $PF\_2Q$.

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