
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
Consider a hyperbola given by the equation $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ and a parabola given by the equation $y = \frac{1}{8}x^2$. These two curves share a common focus $F$. The length of the chord on the hyperbola that passes through the focus $F$ and is perpendicular to the real axis is $\frac{2\sqrt{3}}{3}$. Determine the eccentricity of the hyperbola.

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