
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
Let  $O$  be the center of an equilateral triangle  $ABC$  of area  $1/\pi$ .  As shown in the diagram below, a circle centered at  $O$  meets the triangle at points  $D$ ,  $E$ ,  $F$ ,  $G$ ,  $H$ , and  $I$ , which trisect each of the triangle's sides.  Compute the total area of all six shaded regions. 
[asy]
unitsize(90);
pair A = dir(0);
pair B = dir(120);
pair C = dir(240);
draw(A -- B -- C -- cycle);
pair D = (2*A + B)/3;
pair E = (A + 2*B)/3;
pair F = (2*B + C)/3;
pair G = (B + 2*C)/3;
pair H = (2*C + A)/3;
pair I = (C + 2*A)/3;
draw(E -- F);
draw(G -- H);
draw(I -- D);
draw(D -- G);
draw(E -- H);
draw(F -- I);
pair O = (0, 0);
real r = 1/sqrt(3);
draw(circle(O, r));
fill(O -- D -- E -- cycle, gray);
fill(O -- F -- G -- cycle, gray);
fill(O -- H -- I -- cycle, gray);
fill(arc(O, r, -30, 30) -- cycle, gray);
fill(arc(0, r, 90, 150) -- cycle, gray);
fill(arc(0, r, 210, 270) -- cycle, gray);
label(" $A$ ", A, A);
label(" $B$ ", B, B);
label(" $C$ ", C, C);
label(" $D$ ", D, unit(D));
label(" $E$ ", E, unit(E));
label(" $F$ ", F, unit(F));
label(" $G$ ", G, unit(G));
label(" $H$ ", H, unit(H));
label(" $I$ ", I, unit(I));
label(" $O$ ", O, C);
[/asy]




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