
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 $BC$ of triangle $ABC$ with obtuse angle $C$, point $M$ is marked. Point $D$ is chosen such that triangle $BCD$ is acute, and points $A$ and $D$ lie on opposite sides of line $BC$. Circles $\omega_{B}$ and $\omega_{C}$ are described around triangles $BMD$ and $CMD$ respectively. Side $AB$ intersects circle $\omega_{B}$ again at point $P$, and ray $AC$ intersects circle $\omega_{C}$ again at point $Q$. Segment $PD$ intersects circle $\omega_{C}$ again at point $R$, and ray $QD$ intersects circle $\omega_{B}$ again at point $S$. Find the ratio of the areas of triangles $ABR$ and $ACS$.

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