
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
A hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, |AB|, and |OB| form an arithmetic sequence, and the vectors BF and FA are in the same direction. Determine the eccentricity of the hyperbola.

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