
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
The median \( AE \) and the angle bisector \( CD \) of the isosceles triangle \( ABC \) (\( AB = BC \)) intersect at point \( M \). A line passing through point \( M \) parallel to \( AC \) intersects \( AB \) and \( BC \) at points \( P \) and \( Q \) respectively. Find \( MQ \) and the radius of the circumcircle of triangle \( PQB \), given that \( AC = 4 \) and \( \angle ACB = \arctan 2\sqrt{2} \).

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