
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, point $F$ lies on line segment $CD$, and triangles $CFB$ and $DFE$ are isosceles. Also, $\angle EFB$ is three times $\angle CFE$. What is the measure of $\angle DFE$ in degrees? [asy]
import olympiad;
import math;

size(7cm);

// Draw triangles
pair c = (-2 * Cos(50), 0);
pair d = (1, 0);
pair e = dir(70);
pair f = dir(150);
pair b = (0, 0);
draw(c--f--b--cycle);
draw(b--d--e--cycle);

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

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

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