
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
A circle with center at point \(O\) is circumscribed around triangle \(ABC\). A tangent to the circle at point \(C\) intersects the bisector of angle \(B\) of the triangle at point \(K\), such that angle \(BKC\) is equal to half the difference of three times angle \(A\) and angle \(C\) of the triangle. The sum of the sides \(AC\) and \(AB\) is \(2 + \sqrt{3}\), and the sum of the distances from point \(O\) to the sides \(AC\) and \(AB\) is \(2\). Find the radius of the circle.

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