
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 quadrilateral \(ABCD\) is inscribed in a circle with center \(O\). Two circles \(\Omega_1\) and \(\Omega_2\) of equal radii with centers \(O_1\) and \(O_2\) are inscribed in the angles \(BAD\) and \(BCD\) respectively, with the first touching side \(AB\) at point \(L\) and the second touching side \(BC\) at point \(F\).

a) Find the radius of the circle \(\Omega_2\) if \(AL = \sqrt{2}\) and \(CF = 2\sqrt{2}\).

b) Suppose additionally that point \(O_2\) is the circumcenter of triangle \(BOC\). Find the angle \(BDC\).

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