
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 an ellipse $G$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$, its foci and one vertex are on the circle $x^{2} + y^{2} = 4$.

$(1)$ Find the equation of the ellipse;

$(2)$ Given point $P(-3,2)$, if a line $l$ with slope $1$ intersects the ellipse $G$ at points $A$ and $B$, forming an isosceles triangle $ABP$ with $AB$ as the base, find the equation of line $l$.

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