
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, where $F\_1$ and $F\_2$ are the left and right foci of ellipse $C$, respectively. The distance from the focus $F\_1$ to an asymptote of the hyperbola $\frac{x^{2}}{2}-y^{2}=1$ is $\frac{\sqrt{3}}{3}$.

(I) Find the equation of ellipse $C$;
(II) Line $AB$: $y=kx+m(k < 0)$ intersects ellipse $C$ at distinct points $A$ and $B$. A circle with $AB$ as its diameter passes through point $F\_2$, and the distance from the origin $O$ to line $AB$ is $\frac{2\sqrt{5}}{5}$. Find the equation of line $AB$.

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