
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 triangle \( ABC \), there are marked points \( K \) and \( L \) respectively, such that the quadrilateral \( BKLC \) is cyclic. Inside this quadrilateral, a point \( M \) is chosen such that the line \( AM \) is the angle bisector of \( \angle BMC \). The ray \( BM \) intersects the circumcircle of triangle \( AMC \) at point \( P \) again, and the ray \( CM \) intersects the circumcircle of triangle \( AMB \) at point \( Q \) again. Find the ratio of the areas of triangles \( ALP \) and \( AKQ \).

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