
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
In the rectangular coordinate system \(xOy\), \(F_1\) and \(F_2\) are the left and right foci of the hyperbola \(x^2 - \frac{y^2}{b^2} = 1 \, (b > 0)\), respectively. A tangent to the circle \(x^2 + y^2 = 1\) is drawn through the point \(F_1\), which intersects the left and right branches of the hyperbola at points \(A\) and \(B\), respectively. If \(F_2B = AB\), then the value of \(b\) is \(\qquad\).

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