
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\ (a > b > 0)$, and $P$ is a point on the ellipse $E$ such that $PF\_2 \perp x$-axis, and $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = \frac{1}{16}a^2$. The line $l$ passes through $F\_1$ and intersects the ellipse $E$ at points $A$ and $B$. $F\_2$ and points $A$ and $B$ form $\triangle ABF\_2$.

1. Find the eccentricity of the ellipse $E$.
2. If the perimeter of $\triangle F\_1 P F\_2$ is $2 + \sqrt{3}$, find the maximum area of $\triangle ABF\_2$.

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