
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
In triangle $ABC$, circles with centers $O_1$ and $O_2$ of equal radius are inscribed respectively in vertices $C$ and $A$. Point $O$ is the center of the circle inscribed in triangle $ABC$. These circles touch side $AC$ at points $K_1$, $K_2$, and $K$ respectively, with $CK_1 = 6$, $AK_2 = 8$, and $AC = 21$.

a) Find the length of segment $CK$.

b) Let the circle with center $O_1$ touch side $BC$ at point $K_3$. Find the angle $BCA$ if it is known that $O_1$ is the center of the circle circumscribed around triangle $OK_1K_3$.

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