
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 triangle $PQR$ shown below, $PR=72$. The point $S$ lies on segment $PR$ such that $S$ bisects angle $Q$.
$PQ = 32$ and $QR = 64$. Find the length of $PS$.

[asy]
import markers;

real t=.67;
pair P=(0,0);
pair R=(3,-2);
pair Q=(1.5,1.5);
pair S=t*P+(1-t)*R;

draw(Q--P--R--Q--S);

label("$P$",P,SW);
label("$R$",R,E);
label("$Q$",Q,N);
label("$S$",S,SW);

markangle(n=1,radius=15,P,Q,S,marker(markinterval(stickframe(n=1),true)));
markangle(n=1,radius=15,S,Q,R,marker(markinterval(stickframe(n=1),true)));

label("$64$",.5*(R+Q),E);
label("$32$",.5*(P+Q),NW);
label("$72$",.5*(P+R),S);

[/asy]

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