
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 points on the plane \( A_{\mathrm{t}}=(1+t, 1+t) \) and \( B_{\mathrm{t}}=(-1+t, 1-t) \), describe the set swept by all lines \( A_{\mathrm{t}} B_{\mathrm{t}} \) for all real numbers \( t \).

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