
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
Triangle  $ABC$  has sidelengths  $AB=1$ ,  $BC=\sqrt{3}$ , and  $AC=2$ . Points  $D,E$ , and  $F$  are chosen on  $AB, BC$ , and  $AC$  respectively, such that  $\angle EDF = \angle DFA = 90^{\circ}$ . Given that the maximum possible value of  $[DEF]^2$  can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $\gcd (a, b) = 1$ , find  $a + b$ . (Here  $[DEF]$  denotes the area of triangle  $DEF$ .) 

*Proposed by Vismay Sharan*

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