
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 more than three triangles. If each triangle has the same probability of being selected, what is the probability that a selected triangle has at least one vertex in the shaded region? Express your answer as a common fraction.

[asy]
draw((0,0)--(2,0)--(1,2)--(0,0)--cycle,linewidth(1));
draw((0,0)--(1,0)--(1,1)--(0,0)--cycle,linewidth(1));

label("A",(0,0),SW);
label("B",(1,2),N);
label("C",(2,0),SE);
label("D",(1,0),S);
label("E",(1,1),NE);

filldraw((1,0)--(2,0)--(1,1)--(1,0)--cycle,gray,black);[/asy]

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