
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
A circle $C_{1}$ with a radius of $2\sqrt{6}$ and center $O_{1}$, and a circle $C_{2}$ with a radius of $\sqrt{6}$ and center $O_{2}$ are positioned such that $O_{1}O_{2} = \sqrt{70}$. Line $l_{1}$ is tangent to both circles at points $A_{1}$ and $A_{2}$, and line $l_{2}$ is tangent to both circles at points $B_{1}$ and $B_{2}$. Circles $C_{1}$ and $C_{2}$ lie on the same side of line $l_{1}$ and on opposite sides of line $l_{2}$. Points $A_{1}$ and $B_{1}$ belong to $C_{1}$, and points $A_{2}$ and $B_{2}$ belong to $C_{2}$. Points $A_{1}$ and $B_{1}$ lie on opposite sides of the line segment $O_{1}O_{2}$. A line $l_{3}$ is drawn through point $B_{2}$, perpendicular to line $l_{2}$. Line $l_{1}$ intersects line $l_{2}$ at point $A$, and intersects line $l_{3}$ at point $B$. Find the lengths $A_{1}A_{2}$, $B_{1}B_{2}$, and the sides of triangle $ABB_{2}$.

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