
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
Circle \( C_{1} \) with radius \( 2 \sqrt{3} \) centered at \( O_{1} \) and Circle \( C_{2} \) with radius \( \sqrt{3} \) centered at \( O_{2} \) are positioned such that \( O_{1} O_{2} = 2 \sqrt{13} \). Line \( l_{1} \) is tangent to the circles at points \( A_{1} \) and \( A_{2} \), and line \( l_{2} \) is tangent at points \( B_{1} \) and \( B_{2} \). Circles \( C_{1} \) and \( C_{2} \) lie on one side of line \( l_{1} \) and on different sides of line \( l_{2} \). Points \( A_{1} \in C_{1} \), \( B_{1} \in C_{1} \), \( A_{2} \in C_{2} \), and \( B_{2} \in C_{2} \), with points \( A_{1} \) and \( B_{1} \) on opposite sides of line \( O_{1} O_{2} \). A line \( l_{3} \) through point \( B_{1} \) is perpendicular to line \( l_{2} \). Line \( l_{1} \) intersects line \( l_{2} \) at point \( A \) and intersects line \( l_{3} \) at point \( B \). Find \( A_{1} A_{2} \), \( B_{1} B_{2} \), and the sides of triangle \( A B B_{1} \).

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