
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
In the diagram, lines $\overline{AD} \| \overline{FG} \| \overline{HI}$, point $E$ is on line $\overline{HI}$, and $\overline{EFG}$ forms a straight line. The angle $\angle FHI$ is given as $2x$, and $\angle FEG$ is given as $\cos^{-1}(\frac{1}{3})$. Find the measure of angle $EFG$.

[asy]
import olympiad;

pair A = (-15,20);
pair B = (-12,35);
pair C = (35,50);
pair D = (35,20);
pair E = (14,0);
pair F = (0,0);
pair G = (40,0);
pair H = (0,20);
pair I = (40,20);

draw(F--G);
draw(F--I);
draw(A--D);
draw(H--I);
draw(B--E);

label("F", F, SW);
label("G", G, SE);
label("H", H, NW);
label("I", I, NE);
label("E", E, S);
label("A", A, W);
label("D", D, E);

draw(scale(20)*anglemark(G, F, I));
draw(shift(E)*scale(35)*shift(-E)*anglemark(B, E, A));
draw(shift(E)*scale(20)*shift(-E)*anglemark(I, E, H));

label("$\cos^{-1}(\frac{1}{3})$", (13,0), S);
label("$2x$", (5,20), NW);
[/asy]

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