
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, there are five triangles formed by points A, B, C, D, E, and F. Calculate the probability that a selected triangle has all or part of its interior shaded. Express your answer as a common fraction.

[asy]
draw((0,0)--(1,0)--(0,1)--cycle, linewidth(1));  // Triangle ABC
draw((1,0)--(2,0)--(1,1)--cycle, linewidth(1));  // Triangle CDF
draw((0,0)--(1,0)--(0.5,0.5)--cycle, linewidth(1)); // Triangle AED
draw((1,0)--(1.5,0.5)--(2,0)--cycle, linewidth(1)); // Triangle DFC
draw((0,0)--(0.5,0.5)--(1,0)--cycle, linewidth(1)); // Triangle ADE

label("A", (0,1), NW);
label("B", (0.5,0.5), N);
label("C", (1,0), S);
label("D", (1.5,0.5), N);
label("E", (2,0), S);
label("F", (1,1), NE);

filldraw((0,0)--(0.5,0.5)--(1,0)--cycle, gray, black); // Shade Triangle ADE
filldraw((1,0)--(1.5,0.5)--(2,0)--cycle, gray, black); // Shade Triangle DFE
[/asy]

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