
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 the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{5} = 1$ with left and right foci $F\_1$ and $F\_2$, a chord $AB$ passes through $F\_1$. If the length of the incircle of triangle $ABF\_2$ is $2π$, and the coordinates of points $A$ and $B$ are $(x\_1, y\_1)$ and $(x\_2, y\_2)$, find the value of $|y\_2 - y\_1)|$.

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