
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 a point $M(2 \sqrt {2}, \frac {2 \sqrt {3}}{3})$ on the ellipse $G$: $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a > b > 0)$, and the sum of the distances from point $M$ to the two foci is $4 \sqrt {3}$.  
$(1)$ Find the equation of the ellipse $G$;  
$(2)$ If a line $l$ with a slope of $1$ intersects the ellipse $G$ at points $A$ and $B$, and an isosceles triangle is formed with $AB$ as the base and the vertex at $P(-3,2)$, find the area of $\triangle PAB$.

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