
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 \( \triangle ABC \), \( AC > AB \). Let \( P \) be the intersection point of the perpendicular bisector of \( BC \) and the angle bisector of \( \angle A \). Construct \( PX \perp AB \), intersecting the extension of \( AB \) at point \( X \), and \( PY \perp AC \), intersecting \( AC \) at point \( Y \). Let \( Z \) be the intersection point of \( XY \) and \( BC \). Find the value of \( \frac{BZ}{ZC} \).

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