
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 $A A_{1}, B B_{1}, C C_{1}$ of the exterior angles of triangle $A B C$ intersect the extensions of the opposite sides of the triangle at points $A_{1}, B_{1}, C_{1}$ respectively. Find the angle $A_{1} C_{1} B_{1}$ and the length of segment $A_{1} B_{1}$, given that $A C = 5$, $B C = 2$, and the angle $A C B$ is equal to $\arccos \frac{13}{20}$. (16 points)

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