
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
Let  $\vartriangle ABC$  be an equilateral triangle with side length  $M$  such that points  $E_1$  and  $E_2$  lie on side  $AB$ ,  $F_1$  and  $F_2$  lie on side  $BC$ , and  $G1$  and  $G2$  lie on side  $AC$ , such that   $$ m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2} $$   and the area of polygon  $E_1E_2F_1F_2G_1G_2$  equals the combined areas of  $\vartriangle AE_1G_2$ ,  $\vartriangle BF_1E_2$ , and  $\vartriangle CG_1F_2$ . Find the ratio  $\frac{m}{M}$ .
![Image](https://cdn.artofproblemsolving.com/attachments/a/0/88b36c6550c42d913cdddd4486a3dde251327b.png)

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