
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 three lines on the plane: \(a\), \(o\), and \(c\). Consider the squares \(ABCD\) such that vertex \(A\) is on line \(a\), the opposite vertex \(C\) is on line \(c\), and the center \(O\) of the square lies on line \(o\). Determine the locus of vertices \(B\) and \(D\).

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