
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 a diagram, point $F$ lies on line segment $CD$, and triangles $CFE$ and $DFG$ are isosceles. Furthermore, $\angle EFG$ is three times $\angle CFE$. Determine the measure of $\angle DGF$ in degrees, if $\angle CFE = 50^\circ$. Here is a hypothetical rendering for clarity:
[asy]
import olympiad;
import math;

size(7cm);

// Draw triangles
pair c = (-2 * Cos(50), 0);
pair d = (1, 0);
pair g = dir(20); 
pair e = dir(130);
pair f = (0, 0);
draw(c--f--e--cycle);
draw(f--d--g--cycle);

// Labels
label("$C$", c, W); label("$D$", d, E);
label("$G$", g, N); label("$E$", e, N);
label("$F$", f, S); label("$50^\circ$", c, 2 * NE + 0.5 * E);

// Ticks
add(pathticks(c--f, s=2));
add(pathticks(f--e, s=2));
add(pathticks(f--d, 2, spacing=0.7, s=2));
add(pathticks(f--g, 2, spacing=0.7, s=2));
[/asy]

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