
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
Given two circles \( k_{1} \) and \( k_{2} \) that intersect at the different points \( P \) and \( Q \). Construct a line \( AB \) running through \( P \) with its endpoints at \( k_{1} \) and \( k_{2} \) so that the product \( |AP| \cdot |PB| \) is maximal.

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