
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
Circles with centers \( O_1 \) and \( O_2 \) and equal radii are inscribed in the angles at \( B \) and \( C \) of triangle \( ABC \) respectively. Point \( O \) is the center of the incircle of triangle \( ABC \). These circles touch side \( BC \) at points \( K_1, K_2, \) and \( K \) respectively, with \( BK_1 = 4, CK_2 = 8, \) and \( BC = 18 \).

a) Find the length of segment \( CK \).

b) Suppose the circle with center \( O_1 \) touches side \( AB \) at point \( K_3 \). Find the angle \( ABC \), given that point \( O_1 \) is the center of the circumscribed circle around triangle \( OK_1K_3 \).

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