
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 an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$, a perpendicular line to the x-axis passing through the left focus intersects the ellipse at points $A$ and $B$, and $|AB| = 1$. (1) Find the equation of ellipse $E$. (2) Let $P$ and $Q$ be two points on the ellipse $E$, with $P$ in the first quadrant and $Q$ in the second quadrant, and $OP \perp OQ$, where $O$ is the origin. When $P$ and $Q$ move, is there a fixed circle $O$ such that line $PQ$ is tangent to circle $O$? If it exists, find the equation of circle $O$; if not, explain why.

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