
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 the diagram below, triangle $PQR$ has been reflected over line segment $\overline{PR}$ to produce triangle $PQ'S'$. If $PQ = 9$, $QR = 15$, and $PS = 20$, then find the length of $PQ'$.

[asy]
size(250);
pair P,Q,R,S,PP,QQ,SS;
P = (0,0);
Q = (9,0);
R = (4,12);
S = (20,0);
QQ = reflect(P,R)*Q;
SS = reflect(P,R)*S;
PP = P;
draw(P--Q--R--P--S--R);
draw(P--QQ--SS--P);
label("$P$",P,SW);
label("$Q$",Q,SE);
label("$R$",R,N);
label("$S$",S,SE);
label("$Q'$",QQ,NW);
label("$S'$",SS,NE);
[/asy]

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