
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
The diagram below shows twelve  $30-60-90$  triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. The fourth and last triangle in this diagram are shaded. The ratio of the perimeters of these two triangles can be written as  $\tfrac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

[asy]
size(200);
defaultpen(linewidth(0.8));
pair point=(-sqrt(3),0);
pair past,unit;
path line;
for(int i=0;i<=12;++i) {
past = point;
line=past--origin;
unit=waypoint(line,1/200);
point=extension(past,rotate(90,past)*unit,origin,dir(180-30*i));
if (i == 4) {
filldraw(origin--past--point--cycle,gray(0.7));
} else if (i==12) {
filldraw(origin--past--point--cycle,gray(0.7));
} else {
draw(origin--past--point);
}
}
draw(origin--point);
[/asy]

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