
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 the focal length of ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $2$, and line $l$: $y=x+2$ is tangent to the circle $O$ with the origin $O$ as the center and the minor axis of the ellipse as the diameter.
(1) Find the equation of ellipse $C$;
(2) Let the intersection point of ellipse $C$ and line $y=kx(k > 0)$ in the first quadrant be $A$.
$\quad\quad$ $①$ Set $B( \sqrt {2},1)$, and $ \overrightarrow{OA}⋅ \overrightarrow{OB}= \sqrt {6}$, find the value of $k$;
$\quad\quad$ $②$ If $A$ and $D$ are symmetric about the $x$-axis, find the maximum value of the area of $\triangle AOD$.

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