
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 an ellipse \( E: \frac{x^{2}}{8} + \frac{y^{2}}{b^{2}} = 1 \) with foci on the \( x \)-axis and an inscribed circle \( C: x^{2} + y^{2} = \frac{8}{3} \). The tangent line \( l \) to the circle \( C \) intersects the ellipse \( E \) at two points \( A \) and \( B \), such that \( \overrightarrow{O A} \) is perpendicular to \( \overrightarrow{O B} \) (where \( O \) is the origin).

1. Find the value of \( b^{2} \).
2. Determine the range of \( |A B| \).

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