
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^2}{9} + \frac{y^2}{b^2} = 1 (3 > b > 0)$, if there exists a circle passing through the foci $F\_1$ and $F\_2$ that is tangent to the line $x + y + 2 = 0$, find the maximum value of the ellipse's eccentricity.

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