
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 $x^{2}+\frac{y^{2}}{b^{2}}=1(0 < b < 1)$, respectively. A line $l$ passing through $F\_1$ intersects the ellipse at points $A$ and $B$. If $|AF\_1|=3|F\_1B|$ and $AF\_2 \perp x$-axis, calculate $b^{2}$.

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