
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 triangle \( XYZ \), \( E \) lies on \( \overline{XZ} \) and \( G \) lies on \( \overline{XY} \). Let \( \overline{XE} \) and \( \overline{YG} \) intersect at \( Q \).

[asy]
unitsize(0.8 cm);

pair X, Y, Z, E, G, Q;

X = (1,4);
Y = (0,0);
Z = (6,0);
E = interp(Y,Z,5/11);
G = interp(X,Y,3/8);
Q = extension(X,E,Y,G);

draw(X--Y--Z--cycle);
draw(X--E);
draw(Y--G);

label("$X$", X, N);
label("$Y$", Y, SW);
label("$Z$", Z, SE);
label("$E$", E, S);
label("$G$", G, W);
label("$Q$", Q, SW);
[/asy]

If \( XQ:QE = 5:2 \) and \( GQ:QY = 1:3 \), find \( \frac{XG}{GY} \).

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