
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 $DEF$ (hypotenuse $\overline{DE}$) is inscribed in equilateral triangle $XYZ,$ as shown. If $XD = 4$ and $DY = EZ = 3,$ compute $EZ.$

[asy]
unitsize(0.8 cm);

pair D, E, F, X, Y, Z;

X = (0,0);
Y = (6,0);
Z = 6*dir(60);
D = Y + 3*dir(120);
E = 3*dir(60);
F = (4,0);

draw(D--E--F--cycle);
draw(X--Y--Z--cycle);
draw(rightanglemark(D,F,E,10));

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

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