
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 the diagram, there are two triangles with given side lengths in centimeters. Find the length of segment $XY$:

[asy]
pair X, Y, Z, W, V, U;

X=(0,0);
Y=18*dir(0);
Z=27*dir(110);
W=10+Y;
V=W+9*dir(0);
U=W+15*dir(110);

draw(X--Y--Z--cycle);
draw(W--V--U--cycle);

label("X",U,N);
label("Y",V+(1.4,0));
label("9",.5*(X+Y),S);
label("21",.5*(Y+Z),NE);
label("15",.5*(X+Z),NW);
label("\small{$110^{\circ}$}",X,NW);
label("4.5",.5*(W+V),S);
label("7.5",.5*(W+U),NW);
label("\small{$110^{\circ}$}",W+(2,0.6));
[/asy]

Hint: Assume that $\triangle XYZ \sim \triangle WUV$ by AA similarity (Angle-Angle criterion).

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