
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
Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, respectively. A line $l$ passing through the left focus with a slope of $1$ intersects $E$ at points $A$ and $B$, and $AF_{2}$, $AB$, $BF_{2}$ form an arithmetic sequence.  
$(1)$ Find the eccentricity of $E$;  
$(2)$ If points $A$ and $B$ both lie on a circle centered at $P(-2,0)$, find the equation of $E$.

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