
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 the ellipse $C$: $\frac{x^2}{a^2} + y^2 = 1 (a > 1)$, $A(a,0)$, $B(0,1)$, and the center of the circle $O$: $x^2 + y^2 = 1$ is at a distance of $\frac{\sqrt{3}}{2}$ from the line $AB$.  
(I) Find the equation of the ellipse $C$;  
(II) If the line $l$ is tangent to the circle $O$ and intersects the ellipse $C$ at points $P$ and $Q$, find the maximum value of $|PQ|$.

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