
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
An isosceles right triangle $XYZ$ (with hypotenuse $\overline{XY}$) is inscribed inside an equilateral triangle $DEF,$ as shown in the diagram. If $DX = 5$ and $XE = EY = 4,$ compute the length of $ZD.$

[asy]
unitsize(0.8 cm);

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

D = (0,0);
E = (7,0);
F = 7*dir(60);
X = E + 6/5*dir(120);
Y = 4*dir(60);
Z = (5,0);

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

label("$X$", X, NE);
label("$Y$", Y, NW);
label("$Z$", Z, S);
label("$D$", D, SW);
label("$E$", E, SE);
label("$F$", F, N);
label("$4$", (Z + E)/2, S);
label("$5$", (Z + D)/2, S);
label("$4$", (Y + E)/2, NW);
[/asy]

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