
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 an ellipse with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. The right focus of the ellipse coincides with the focus $F$ of the parabola $y^2 = 4x$.

1. Find the equation of the ellipse.
2. A line $l$ passing through $F$ intersects the ellipse at points $A$ and $B$, and the left focus of the ellipse is $F'$. Find the maximum area of triangle $AF'B$.

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