
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 triangle \(ABC\), where \(AB = BC = 3\) and \(\angle ABC = \arccos \frac{1}{9}\), median \(AD\) and angle bisector \(CE\) intersect at point \(M\). A line parallel to \(AC\) is drawn through \(M\), intersecting sides \(AB\) and \(BC\) at points \(P\) and \(Q\) respectively. Find \(PM\) and the radius of the circle inscribed in triangle \(PQB\).

Also, circle \(\sigma\) with center \(O\) on side \(AC\) of triangle \(ABC\) touches sides \(AB\) and \(BC\) at points \(D\) and \(E\) respectively. Given that \(AD = 2CE\), and the angle \(\angle DOE\) is \(\operatorname{arcctg} \frac{1}{3}\). Find the angles of triangle \(ABC\) and the ratio of its area to the area of the circle \(\sigma\).

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