
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 \( A_1 \) and \( B_1 \) divide the sides \( BC \) and \( AC \) of triangle \( ABC \) in the ratios \( BA_1 : A_1C = 1 : p \) and \( AB_1 : B_1C = 1 : q \). In what ratio is the segment \( AA_1 \) divided by the segment \( BB_1 \)?

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