
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 at \(O_1\) and \(O_2\) and equal radii are inscribed at the angles \(C\) and \(B\) of triangle \(ABC\). \(O\) is the center of the circle inscribed in triangle \(ABC\). These circles touch the side \(BC\) at points \(K_1\), \(K_2\), and \(K\) respectively, with \(CK_1 = 3\), \(BK_2 = 7\), and \(BC = 16\).

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

b) Let the circle with center \(O_1\) touch side \(AC\) at point \(K_3\). Find the angle \(ACB\), given 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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