
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 \( ABC \) be a triangle with \( AB = 13 \), \( AC = 14 \), and \( BC = 15 \). Let \( G \) be the point on \( AC \) such that the reflection of \( BG \) over the angle bisector of \( \angle B \) passes through the midpoint of \( AC \). Let \( Y \) be the midpoint of \( GC \) and \( X \) be a point on segment \( AG \) such that \( \frac{AX}{XG} = 3 \). Construct \( F \) and \( H \) on \( AB \) and \( BC \), respectively, such that \( FX \parallel BG \parallel HY \). If \( AH \) and \( CF \) concur at \( Z \) and \( W \) is on \( AC \) such that \( WZ \parallel BG \), find \( WZ \).

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