
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
A circle has an equilateral triangle and a rectangle inscribed in it. They share a common vertex. The rectangle has one angle of 60 degrees. Find the measure of the angle indicated by the question mark at the shared vertex.

[asy]
import markers; defaultpen(linewidth(0.8));

pair A,B,C,D,E,F,G;

draw(unitcircle);

A=(-1,0);
B=(0,-1);
C=(1,0);
D=(0,1);

draw(A--B--C--D--A);

E=(-0.9,-0.45);
F=(0.9,-0.45);

draw(D--E--F--D);

G=(-0.76,-0.23);

markangle(Label("?"),n=1,radius=10,D,G,A,marker(stickframe(n=0),true));

label("$A$",(-0.5,-0.34));
label("$B$",(0.5,-0.34));
label("$C$",B,S);

[/asy]

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