
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 points $F_1$ and $F_2$ are respectively the left and right foci of the ellipse $\frac{x^2}{k+2} + \frac{y^2}{k+1} = 1 (k > -1)$. Chord $AB$ passes through point $F_1$, and if the perimeter of $\triangle ABF_2$ is $8$, calculate the eccentricity of the ellipse.

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