
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 that the vertex of the parabola $C_1$ is at the origin $O$ and it passes through point $M(\frac{2}{3}, \frac{2\sqrt{6}}{3})$, determine the eccentricity of the hyperbola $C_2$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\ (a > 0, b > 0)$.

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