
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 point $P(-1, \frac{3}{2})$ is a point on the ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, $F_{1}$, $F_{2}$ are the left and right foci of the ellipse $E$ respectively, $O$ is the coordinate origin, and $PF_{1} \perp x$ axis.
(1) Find the equation of the ellipse $E$;
(2) Given that circle $O$: $x^{2}+y^{2}=r^{2}(0 < r < b)$, line $l$ is tangent to circle $O$ and intersects the ellipse at points $A$ and $B$. If $\overrightarrow{OA}\cdot \overrightarrow{OB}=0$, find the equation of circle $O$.

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