
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
On the sides \(AB\) and \(AC\) of an acute-angled triangle \(ABC\), points \(K\) and \(L\) are marked such that the quadrilateral \(BKLC\) is inscribed. Inside this quadrilateral, a point \(M\) is chosen such that the line \(AM\) is the bisector of angle \(BMC\). Ray \(BM\) intersects the circumcircle of triangle \(AMC\) at point \(P\) for the second time, and ray \(CM\) intersects the circumcircle of triangle \(AMB\) at point \(Q\) for the second time. Find the ratio of the areas of triangles \(ALP\) and \(AKQ\).

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