
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 $F_1$ and $F_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) respectively, and $|F_1F_2| = 2$. A line $l$ passes through $F_1$ and is perpendicular to the $x$-axis, intersecting the ellipse $C$ at points $A$ and $B$. Connecting $A$, $B$, and $F_2$ forms an equilateral triangle.

(I) Find the equation of the ellipse $C$.
(II) A line $m$ passing through the right focus $F_2$ intersects the ellipse $C$ at points $M$ and $N$. Determine whether there exists a point $P$ on the ellipse $C$ such that $\overrightarrow{OP} = \overrightarrow{OM} + \overrightarrow{ON}$ holds. If it exists, find the coordinates of point $P$; if not, explain the reason.

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