
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 $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1({a＞b＞0})$, a circle with center at the origin $O$ and radius $\sqrt{{a^2}+{b^2}}$ is called the "pseudo-circle" of ellipse $C$. It is known that one focus of ellipse $C$ is $F({\sqrt{2},0})$, and the distance from one endpoint of its minor axis to point $F$ is $\sqrt{3}$.  
$(1)$ Find the equations of ellipse $C$ and its "pseudo-circle";  
$(2)$ If point $A$ is the intersection point of the "pseudo-circle" of ellipse $C$ and the positive $x$-axis, $B$ and $D$ are two distinct points on ellipse $C$, and $BD\bot x$ axis, find the range of values of $\overrightarrow{AB}•\overrightarrow{AD}$.

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