
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
Right triangle $ABC$ (hypotenuse $\overline{AB}$) is inscribed in equilateral triangle $XYZ,$ as shown. If $XC = 4$ and $BX = CZ = 3,$ compute $AZ$.

[asy]
unitsize(0.8 cm);

pair A, B, C, X, Z, Y;

X = (0,0);
Z = (6,0);
Y = 6*dir(60);
A = Z + 1.2*dir(120);
B = 3*dir(60);
C = (4,0);

draw(A--B--C--cycle);
draw(X--Z--Y--cycle);
draw(rightanglemark(A,C,B,10));

label("$A$", A, NE);
label("$B$", B, NW);
label("$C$", C, S);
label("$X$", X, SW);
label("$Z$", Z, SE);
label("$Y$", Y, N);
label("$3$", (C + Z)/2, S);
label("$4$", (C + X)/2, S);
label("$3$", (B + X)/2, NW);
[/asy]

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