
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, line $AB$ is parallel to line $EF$ and $AFC$ is a straight line. Points $D$, $E$ are on line $AB$, and $G$ is on $EF$. Given that $\angle AFC = 180^\circ$, $\angle AFE = 100^\circ$, and $\angle EFA = 60^\circ$, determine the value of $x,$ which is $\angle DGC$.
[asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)--cycle);
draw((-.5,5)--(8.5,-1)); // Extended line $FC$
label("$A$",(-.5,5),W);
label("$B$",(8,5),E);
label("$C$",(6.5,0),S);
label("$D$",(0,0),SW);
label("$F$",(8,8),E);
label("$E$", (4,8), N);
label("$G$", (6.5,8), N);
draw((2,0)--(3,0),arrow = Arrow);
draw((3,0)--(4,0),arrow = Arrow);
draw((2,5)--(3,5),arrow = Arrow);
draw((3,5)--(4,5),arrow = Arrow);
label("$x^\circ$", (0.1,4));
label("$100^\circ$", (0,0), NE);
label("$60^\circ$", (8,5), SW);
label("$180^\circ$", (6.5,0), E);
[/asy]

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