
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 $\frac{1}{2}$, and a circle $O$: $x^{2}+y^{2}=r^{2}$ $(r > 0)$ intersects the x-axis at points $M$ and $N$. Let $P$ be a moving point on the ellipse $E$, and $|PM|+|PN|=2a$. The maximum area of $\triangle PMN$ is $\sqrt{3}$.  
$(I)$ Find the equations of circle $O$ and ellipse $E$;  
$(II)$ The tangent line $l$ of circle $O$ intersects the ellipse $E$ at points $A$ and $B$. Find the range of $|AB|$.

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