
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 new diagram, a point F is added to create more triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has all or part of its interior shaded? Modify the diagram as follows, with an additional filled region:

[asy]
draw((0,0)--(2,0)--(0,2)--(0,0)--cycle,linewidth(1));  // Big triangle AFG
draw((0,0)--(1,0)--(1,1)--(0,0)--cycle,linewidth(1));  // Smaller triangle AED
draw((1,0)--(2,0)--(1,1)--cycle, linewidth(1));  // Smaller triangle DFC
label("A",(0,2),NW);
label("E",(.6,.6),NE);
label("D",(1,0),S);
label("F",(2,0),SE);
label("G",(0,0),SW);
filldraw((1,0)--(2,0)--(1,1)--cycle, gray, black);  // fill triangle DFC
filldraw((0,0)--(1,0)--(.5,.5)--cycle, gray, black); // fill triangle AED part
[/asy]

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