
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 $ABC,$ $G$ lies on $\overline{AB}$ such that $AG:GB = 3:2,$ and $H$ lies on $\overline{BC}$ such that $BH:HC = 3:1.$ Let $Q$ be the intersection of $\overline{AG}$ and $\overline{CH}.$

[asy]
unitsize(0.8 cm);

pair A, B, C, G, H, Q;

A = (1,4);
B = (0,0);
C = (6,0);
G = interp(A,B,3/5);
H = interp(B,C,3/4);
Q = extension(A,G,C,H);

draw(A--B--C--cycle);
draw(A--G);
draw(C--H);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$G$", G, NW);
label("$H$", H, E);
label("$Q$", Q, S);
[/asy]

Then
\[\overrightarrow{Q} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\]where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$  Find the ordered triple $(x,y,z).$

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