
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: $\frac{x^{2}}{9} + \frac{y^{2}}{b^{2}} = 1 (0 < b < 3)$, with left and right foci $F\_1$ and $F\_2$, respectively. A line $l$ passing through $F\_1$ intersects the ellipse at points $A$ and $B$. Find the value of $b$ such that the maximum value of $|BF_{2}| + |AF_{2}|$ is $10$.

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