
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
In the angle \( A_{1}'OA_{2}' \) formed by two intersecting lines \( a_{1} \) and \( a_{2} \), from an arbitrary point \( P \) inside this angle, perpendiculars \( P B_{1} \) and \( P B_{2} \) are dropped to the lines \( a_{1} \) and \( a_{2} \), respectively. Given two positive numbers \( k_{1} \) and \( k_{2} \), and a given segment length \( d \), find the locus of points \( P \) (lying within the given angle \( A_{1}'OA_{2}' \)) for which

$$
k_{1} \cdot P B_{1} + k_{2} \cdot P B_{2} = d.
$$

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