
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 an ellipse with an eccentricity of $\frac{1}{2}$ and the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If the right focus of this ellipse coincides with the right focus of the hyperbola $x^2 - \frac{y^2}{3} = 1$, find the equation of the ellipse.

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