
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 side lengths of both triangles are given in centimeters. What is the length of segment $PQ$?

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

P = (0,0);
Q = 15*dir(0);
R = 25*dir(135);
S = 12+Q;
T = S+9*dir(0);
U = S+15*dir(135);

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

label("P", U, N);
label("Q", T+(1.6,0));
label("9", .5*(P+Q), S);
label("20", .5*(Q+R), NW);
label("15", .5*(P+R), SW);
label("\small{$135^{\circ}$}", P, NW);
label("4.5", .5*(S+T), S);
label("7.5", .5*(S+U), SW);
label("\tiny{$135^{\circ}$}", S+(2, 1));
[/asy]

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