
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 game board is constructed by shading a sector of a circle inscribed in one of the regions formed by the altitudes of an equilateral triangle and one triangular region, as shown. The circular sector has a central angle of 60 degrees. What is the probability that the tip of the spinner will come to rest in a shaded region? Assume the spinner can land anywhere in the equilateral triangle.
[asy]
import olympiad; size(100); defaultpen(linewidth(0.8));
pair A=(0,0), B=(1,0), C=(0.5,sqrt(3)/2);
pair D=(A+B)/2, E=(B+C)/2, F=(C+A)/2;
pair M=intersectionpoint(A--E,B--F);
draw(A--B--C--cycle);
draw(A--E^^B--F^^C--D);
filldraw(arc(D, M, B, CCW)--D--cycle, gray(0.6));  // Circular sector shaded
filldraw(F--M--C--cycle, gray(0.6));  // Triangular region shaded
draw(M--(0.4,0.5),EndArrow(size=10));
[/asy]

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