
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, medians BD and CF intersect at G (the centroid of triangle ABC), and the midpoint of line segment BF is H. Let the area of triangle HGF be \( j \) times the area of triangle ABC. Determine the value of \( j \).
A) \( \frac{1}{18} \)
B) \( \frac{1}{12} \)
C) \( \frac{1}{6} \)
D) \( \frac{1}{24} \)

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