
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 the right focus of ellipse $I$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > 0,b > 0)$ is $(2 \sqrt{2},0)$, and ellipse $I$ passes through point $(3,1)$.
(1) Find the equation of ellipse $I$;
(2) Let line $l$ with slope $1$ intersect ellipse $I$ at two distinct points $A$ and $B$. Construct an isosceles triangle $PAB$ with base $AB$ and vertex $P$ at coordinates $(-3,2)$. Find the area of $∆PAB$.

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