
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 the ellipse $C: \frac{x^{2}}{4}+y^{2}=1$, and a moving line $l$ with a slope of $\frac{\sqrt{3}}{2}$. The line $l$ intersects with the ellipse $C$ at two distinct points $A$ and $B$.

1. Let $M$ be the midpoint of the chord $AB$, find the trajectory equation of the moving point $M$.
2. Let $F\_1$ and $F\_2$ be the left and right foci of the ellipse $C$, and let $P$ be a point on the ellipse in the first quadrant such that $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = -\frac{5}{4}$. Find the maximum area of $\triangle PAB$.

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