
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
Quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Two circles \(\Omega_{1}\) and \(\Omega_{2}\) of equal radii and centers \(O_{1}\) and \(O_{2}\) respectively are inscribed in angles \(ABC\) and \(ADC\). The first circle is tangent to side \(BC\) at point \(K\), and the second circle is tangent to side \(AD\) at point \(T\).

a) Find the radius of the circle \(\Omega_{1}\), given that \(BK = 3\sqrt{3}\) and \(DT = \sqrt{3}\).

b) Additionally, suppose that point \(O_{1}\) is the center of the circle circumscribed around triangle \(BOC\). Find the angle \(BDC\).

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