
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 parallel lines $e_{1}$ and $e_{2}$ in the plane, and a line $f$ perpendicular to them that intersects $e_{1}$ at $O_{1}$ and $e_{2}$ at $O_{2}$, determine the locus of points through which a line can be drawn such that the points of intersection with $e_{1}$ and $e_{2}$, denoted $P_{1}$ and $P_{2}$ respectively, satisfy the following conditions:
a) $O_{1} P_{1} \cdot O_{2} P_{2} = 1$
b) $P_{1}$ and $P_{2}$ are on the same side of $f$.

Also solve the problem assuming that in condition b), "same" is replaced with "opposite".

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