
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
The sides of two triangles are given in meters. Calculate the length of segment $PQ$.

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

P=(0,0);
Q=15*dir(0);
R=25*dir(120);
S=10+Q;
T=S+9*dir(0);
U=S+15*dir(120);

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

label("P",U,N);
label("Q",T+(1.4,0));
label("9",.5*(P+Q),S);
label("21",.5*(Q+R),NE);
label("15",.5*(P+R),SW);
label("\small{$120^{\circ}$}",P,NE);
label("4.5",.5*(S+T),S);
label("7.5",.5*(S+U),SW);
label("\tiny{$120^{\circ}$}",S+(1.8,0.8));

[/asy]

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