
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 eccentricity $\frac{\sqrt{2}}{2}$, where $F_1$ and $F_2$ are the left and right foci of ellipse C, respectively. The distance from the focus $F_1$ of ellipse C 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) The line AB: $y = kx + m$ ($k < 0$) intersects ellipse C at two distinct points A and B. The circle with diameter AB 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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