
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
Points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on the sides of triangle \( ABC \), dividing its sides in the ratios \( B A_1 : A_1 C = p \), \( C B_1 : B_1 A = q \), and \( A C_1 : C_1 B = r \). The points of intersection of the segments \( AA_1 \), \( BB_1 \), and \( CC_1 \) are situated as shown in the diagram. Find the ratio of the areas of triangles \( PQR \) and \( ABC \).

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