
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
A series of lines \( l_{1}, l_{2}, \cdots \) each pass through the focus of the parabola \( C: y^{2} = x \) in the \( xOy \) plane. Each line \( l_{i} \) (for \( i \geqslant 1 \)) intersects the parabola \( C \) at points \( A_{i} \) and \( B_{i} \). If the slope of \( l_{1} \) is \( 1 \) and the slope of \( l_{i} \) (\( i \geqslant 2 \)) is \( 1 + \sqrt{\left|A_{i-1} B_{i-1}\right| - 1} \), determine the equation of \( l_{2014} \).

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