
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 a hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0, b > 0)$, one of its asymptotes intersects with the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$ at a point whose projection on the $x$-axis is exactly a focus of the ellipse. What is the eccentricity of the hyperbola?

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