
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
Stuart has drawn a pair of concentric circles, as shown, and starts drawing chords $\overline{AB}$, $\overline{BC}, \ldots$ of the large circle, each tangent to the small one. If $m\angle ABC=60^\circ$, then how many segments will he draw before returning to his starting point at $A$? 
[asy]
size(100); defaultpen(linewidth(0.8));
real rad1 = 1/Sin(30);
draw(Circle(origin,1)); draw(Circle(origin,rad1));
pair A = rad1*dir(160), B = rad1*dir(160 + 120), C = rad1*dir(160 + 2*120), D = rad1*dir(160 + 300);
draw(A--B--C--D,EndArrow(size=5));
label("$A$",A,W); label("$B$",B,N); label("$C$",C,E);
[/asy]

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