
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
In the Cartesian coordinate plane $xOy$, an ellipse $(E)$ has its center at the origin, passes through the point $A(0,1)$, and its left and right foci are $F_{1}$ and $F_{2}$, respectively, with $\overrightarrow{AF_{1}} \cdot \overrightarrow{AF_{2}} = 0$.

(I) Find the equation of the ellipse $(E)$;

(II) A line $l$ passes through the point $(-\sqrt{3}, 0)$ and intersects the ellipse $(E)$ at exactly one point $P$. It also tangents the circle $(O): x^2 + y^2 = r^2 (r > 0)$ at point $Q$. Find the value of $r$ and the area of $\triangle OPQ$.

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