
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
Let the parabola \( C: y^2 = 2x \) have its directrix intersect the \( x \)-axis at point \( A \). A line \( l \) through point \( B(-1, 0) \) is tangent to the parabola \( C \) at point \( K \). Draw a line through point \( A \) parallel to \( l \), which intersects the parabola \( C \) at points \( M \) and \( N \). Find the area of triangle \( \triangle KMN \).

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