
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
The angle bisectors $A A_{1}$ and $B B_{1}$ of triangle $A B C$ intersect at point $O$, with $A O: O A_{1} = 2: 1$. The angle bisector of the exterior angle at vertex $C$ of triangle $A B C$ intersects line $A B$ at point $C_{1}$. Find the angle $B_{1} A_{1} C_{1}$ and the length of segment $A_{1} C_{1}$ given that $A B = 2$ and $A C = 4$. (16 points)

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