
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
Square $PQRS$ is constructed along diameter $PQ$ of a semicircle. Line segment $PQ$ has a length of 8 centimeters. If point $N$ is the point on the arc $PQ$ such that the angle $\angle PNQ = 60^\circ$, what is the length of segment $NC$? Assume $C$ is the midpoint of side $QR$ of the square. [asy]
size(4cm);

dotfactor = 4;
defaultpen(linewidth(1)+fontsize(10pt));

pair P,Q,R,S,C,N;
P = (0,1);
Q = (1,1);
R = (1,0);
S = (0,0);
N = (.5,1.5);
C = (.5,0);

draw(P..N..Q--R--S--cycle);
draw(P--Q);

dot("P",P,W);
dot("N",N,N);
dot("Q",Q,E);
dot("R",R,E);
dot("S",S,W);
dot("C",C,NW);

draw(N--C,linetype("0 4"));
[/asy]

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