
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 side \( AB \) of an acute-angled triangle \( ABC \), point \( M \) is marked. Inside the triangle, point \( D \) is chosen. Circles \( \omega_{A} \) and \( \omega_{B} \) are circumscribed around triangles \( AMD \) and \( BMD \), respectively. Side \( AC \) intersects circle \( \omega_{A} \) a second time at point \( P \), and side \( BC \) intersects circle \( \omega_{B} \) a second time at point \( Q \). Ray \( PD \) intersects circle \( \omega_{B} \) a second time at point \( R \), and ray \( QD \) intersects circle \( \omega_{A} \) a second time at point \( S \). Find the ratio of the areas of triangles \( ACR \) and \( BCS \).

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