
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 $PQR$ is reflected over its median $\overline{PM}$ to produce triangle $PQ'R'$. If $PE = 8$, $ER =16$, and $QD = 15$, then find $PQ$.

[asy]
size(250);
pair P,Q,R,D,M,QQ,RR,EE;
Q = (0,0);
D = (15,0);
M = (20,0);
R=2*M;
P = D + (scale(1.5)*rotate(aCos((300-169-36)/150))*(M-D));
RR = D + D + D - P - P;
QQ = reflect(P,M)*Q;
EE = reflect(P,M)*D;
draw(M--P--QQ--RR--P--Q--R--P);
label("$M$",M,SE);
label("$P$",P,N);
label("$Q$",Q,SW);
label("$R$",R,SE);
label("$R'$",RR,S);
label("$Q'$",QQ,E);
label("$D$",D,NW);
label("$E$",EE,N);
label("$16$",(EE+R)/2,N);
label("$8$",(P+EE)/2,S);
label("$15$",D/2,S);
[/asy]

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