
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
On equilateral  $\triangle{ABC}$  point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J which are the vertices of another equilateral triangle. The area of  $\triangle{ABC}$  is twice the area of  $\triangle{GHJ}$ . The side length of  $\triangle{ABC}$  can be written  $\frac{r+\sqrt{s}}{t}$ , where r, s, and t are relatively prime positive integers. Find  $r + s + t$ .

[center]![Image](https://i.snag.gy/TKU5Fc.jpg)[/center]

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