
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
In triangle ABC, altitudes BD and CF intersect at point H. The midpoint of AC is P. Let the area of triangle HPC be \( l \) times the area of triangle ABC. Compute the value of \( l \).
A) \( \frac{1}{5} \)    
B) \( \frac{1}{4} \)    
C) \( \frac{1}{3} \)    
D) \( \frac{1}{2} \)

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