
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F\_1$, $F\_2$, respectively. The length of the major axis is $4$, and the eccentricity is $\frac{1}{2}$. A line $AB$ passes through focus $F\_1$ and intersects the ellipse $C$ at points $A$ and $B$ (with $A$ in the first quadrant), and the area ratio of triangles $\triangle F\_1AF\_2$ and $\triangle F\_1BF\_2$ is $7:3$.

1. Find the equation of the ellipse.
2. Find the equation of line $AB$.

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