
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  $AC > AB$ .  Let  $P$  be the intersection point of the perpendicular bisector of  $BC$  and the internal angle bisector of  $\angle{A}$ .  Construct points  $X$  on  $AB$  (extended) and  $Y$  on  $AC$  such that  $PX$  is perpendicular to  $AB$  and  $PY$  is perpendicular to  $AC$ .  Let  $Z$  be the intersection point of  $XY$  and  $BC$ .
Determine the value of  $\frac{BZ}{ZC}$ .

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