
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 the lines \( l_{1}: A x + B y + C_{1} = 0 \) and \( l_{2}: A x + B y + C_{2} = 0 \) where \( C_{1} \neq C_{2} \). Also given that \( A - B + C_{1} + C_{2} = 0 \), find the equation of the line \( l \) that passes through the point \( H(-1,1) \) and has its midpoint \( M \), of the segment intercepted by the parallel lines \( l_{1} \) and \( l_{2} \), lying on the line \( x - y - 1 = 0\).

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