
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ with left and right foci $F_1$ and $F_2$, respectively. A line with a slope of 45° is drawn through $F_1$, intersecting the ellipse at points A and B, and $\overrightarrow{F_1B} = 2\overrightarrow{AF_1}$. Find the eccentricity of the ellipse.

A) $\frac{\sqrt{3}}{3}$
B) $\frac{\sqrt{3}}{2}$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{\sqrt{2}}{3}$

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