
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
The right focus of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ coincides with the focus of the parabola $y^{2}=2px$, where $a \gt 0$ and $b \gt 0$. The directrix of the parabola intersects the hyperbola at points $A$ and $B$, while the asymptotes of the hyperbola intersect at points $C$ and $D$. Given that $|CD|=\sqrt{2}|AB|$, calculate the eccentricity of the hyperbola.

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