
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. The line through  $A$  tangent to the circumcircle of  $ABC$  intersects line  $BC$  at point  $W$ . Points  $X,Y \neq A$  lie on lines  $AC$  and  $AB$ , respectively, such that  $WA=WX=WY$ . Point  $X_1$  lies on line  $AB$  such that  $\angle AXX_1 = 90^{\circ}$ , and point  $X_2$  lies on line  $AC$  such that  $\angle AX_1X_2 = 90^{\circ}$ . Point  $Y_1$  lies on line  $AC$  such that  $\angle AYY_1 = 90^{\circ}$ , and point  $Y_2$  lies on line  $AB$  such that  $\angle AY_1Y_2 = 90^{\circ}$ . Let lines  $AW$  and  $XY$  intersect at point  $Z$ , and let point  $P$  be the foot of the perpendicular from  $A$  to line  $X_2Y_2$ . Let line  $ZP$  intersect line  $BC$  at  $U$  and the perpendicular bisector of segment  $BC$  at  $V$ . Suppose that  $C$  lies between  $B$  and  $U$ . Let  $x$  be a positive real number. Suppose that  $AB=x+1$ ,  $AC=3$ ,  $AV=x$ , and  $\frac{BC}{CU}=x$ . Then  $x=\frac{\sqrt{k}-m}{n}$  for positive integers  $k$ , $m$ , and  $n$  such that  $k$  is not divisible by the square of any integer greater than  $1$ . Compute  $100k+10m+n$ .

*Proposed by Ankit Bisain, Luke Robitaille, and Brandon Wang*

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