
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 foci and one vertex of the ellipse $G$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ are on the circle $x^{2}+y^{2}=4$.
(1) Find the equation of the ellipse;
(2) Given point $P(-3,2)$, if the line $l$ with a slope of $1$ intersects the ellipse $G$ at points $A$ and $B$, discuss whether an isosceles triangle $ABP$ with base $AB$ exists. If it exists, find the equation of line $l$; if not, explain the reason.

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