
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), respectively. The eccentricity of $E$ is $\frac{\sqrt{2}}{2}$, and the point $(0, 1)$ lies on $E$.

(1) Find the equation of the ellipse $E$.

(2) A line passing through point $F_1$ intersects the ellipse $E$ at points $A$ and $B$, and $\overrightarrow{BF_1} = 2\overrightarrow{F_1A}$. Find the equation of the line $BF_2$.

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