
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 that $F\_1(-c,0)$ and $F\_2(c,0)$ are the left and right foci of the ellipse $G$: $\frac{x^2}{a^2} + \frac{y^2}{4} = 1 (a > 0)$, and point $M$ is a point on the ellipse such that $MF\_2 \perp F\_1F\_2$, and $|MF\_1| - |MF\_2| = \frac{4}{3}a$.

1. Find the equation of ellipse $G$.
2. If a line $l$ with a slope of $1$ intersects ellipse $G$ at points $A$ and $B$, and an isosceles triangle is formed with $AB$ as the base and $P(-3, 2)$ as the apex, find the area of $\triangle PAB$.

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