
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 that $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ are points on and off the line $l$, respectively, and the equation of the line $l$ is $f(x, y) = 0$, derive the equation $f(x, y) - f(x_{1}, y_{1}) - f(x_{2}, y_{2}) = 0$.

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