
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
In parallelogram \(ABCD\), lines \(l_1\) and \(l_2\) are the bisectors of angles \(A\) and \(C\) respectively, while lines \(m_1\) and \(m_2\) are the bisectors of angles \(B\) and \(D\) respectively. The distance between \(l_1\) and \(l_2\) is \(\sqrt{3}\) times smaller than the distance between \(m_1\) and \(m_2\). Find the angle \(BAD\) and the radius of the circle inscribed in triangle \(ABD\) if \(AC=\sqrt{\frac{22}{3}}\) and \(BD=2\).

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