
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 $XYZ$ (hypotenuse $\overline{XY}$) is inscribed in an equilateral triangle $LMN$. If $LZ = 4$ and $MX = NX = 3$, compute $NY$.

[asy]
unitsize(0.8 cm);

pair X, Y, Z, L, M, N;

L = (0,0);
M = (6,0);
N = 6*dir(60);
X = M + 9/4*dir(120);
Y = 3*dir(60);
Z = (4,0);

draw(X--Y--Z--cycle);
draw(L--M--N--cycle);
draw(rightanglemark(X,Z,Y,10));

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

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