
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 left and right foci $F_{1}$ and $F_{2}$ respectively, and upper and lower vertices $B_{1}$ and $B_{2}$ respectively, the right vertex is $A$. The line $A B_{1}$ intersects $B_{2} F_{1}$ at point D. If $2\left|A B_{1}\right|=3\left|B_{1} D\right|$, determine the eccentricity of ellipse $C$.

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