
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. Points  $D,E, F$  are drawn on sides  $AB$ , $BC$ , and  $CA$  respectively such that  $[ADF] = [BED] + [CEF]$  and  $\vartriangle ADF \sim  \vartriangle BED \sim \vartriangle CEF$ . The ratio  $\frac{[ABC]}{[DEF]}$  can be expressed as  $\frac{a+b\sqrt{c}}{d}$  , where  $a$ ,  $b$ ,  $c$ , and  $d$  are positive integers such that  $a$  and  $d$  are relatively prime, and  $c$  is not divisible by the square of any prime. Find  $a + b + c + d$ .
(Here  $[P]$  denotes the area of polygon  $P$ .)

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