
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 $A$ and $D$ are respectively the left and top vertices of the ellipse $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1, (a > b > 0)$, the eccentricity of the ellipse is $e = \dfrac{\sqrt{3}}{2}$, $F_1$ and $F_2$ are the left and right foci of the ellipse, and point $P$ is any point on the line segment $AD$, and the maximum value of $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2}$ is $1$.
1. Find the equation of the ellipse $E$.
2. Determine whether there exists a circle with its center at the origin such that any tangent line to this circle intersects the ellipse $E$ at two points $A$ and $B$, and $OA \perp OB$ ($O$ is the origin). If such a circle exists, find its equation; if not, explain why.

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