
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 \(\Omega_{2}\) with equal radii and centers \(O_{1}\) and \(O_{2}\) are inscribed in the angles \(BAD\) and \(BCD\) respectively. The first circle is tangent to side \(AD\) at point \(K\), and the second circle is tangent to side \(BC\) at point \(T\).

a) Find the radius of circle \(\Omega_{1}\) if \(AK = 2\) and \(CT = 8\).

b) Additionally, suppose that point \(O_{2}\) is the circumcenter of triangle \(BOC\). Find angle \(BDC\).

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