
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 $F_1(-c,0)$ and $F_2(c,0)$ are the left and right foci of the ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $|F_1F_2| = 2\sqrt{3}$, eccentricity $e = \frac{\sqrt{3}}{2}$.
(I) Find the standard equation of the ellipse;
(II) A line $l$ passing through the right focus $F_2$ intersects the ellipse $M$ at points $A$ and $B$.
(1) When the slope of line $l$ is 1, find the area $S$ of $\triangle AF_1B$.
(2) Does there exist a point $P$ on the ellipse such that the quadrilateral $OAPB$ with adjacent sides $OA$ and $OB$ is a parallelogram ($O$ is the coordinate origin)? If it exists, find the coordinates of all points $P$ and the equation of line $l$; if not, explain the reason.

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