
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
Given two triangles, where the side lengths of the smaller triangle are 4 cm, 8 cm, and a 120° angle between them, and the side lengths of the larger triangle are 9 cm and 18 cm with a 120° angle between them, find the length of segment $PQ$, which corresponds to the 4 cm side in the smaller triangle.

[asy]
pair P,Q,R,S,T,U;

P=(0,0);
Q=4*dir(0);
R=8*dir(120);
S=9+Q;
T=S+18*dir(0);
U=S+9*dir(120);

draw(P--Q--R--cycle);
draw(S--T--U--cycle);

label("P",U,N);
label("Q",T+(1.4,0));
label("4",.5*(P+Q),S);
label("8",.5*(Q+R),NE);
label("4",.5*(P+R),SW);
label("\small{$120^{\circ}$}",P,NE);
label("9",.5*(S+T),S);
label("9",(.4*S)+(.6*U),SW);
label("\tiny{$120^{\circ}$}",S+(1.8,0.8));

[/asy]

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