
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 the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$ $(a>b>0)$ with two foci $F_1$ and $F_2$, a circle is drawn with the origin O as its center and $OF_1$ as its radius, intersecting the hyperbola E. If the points of intersection are connected sequentially with $F_1$ and $F_2$ to form a regular hexagon, find the eccentricity of the hyperbola E.

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