
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  $ABC$  be an equilateral triangle, and let  $D$  and  $F$  be points on sides  $BC$  and  $AB$ , respectively, with  $FA=5$  and  $CD=2$ .  Point  $E$  lies on side  $CA$  such that  $\angle DEF = 60^\circ$ .  The area of triangle  $DEF$  is  $14\sqrt{3}$ .  The two possible values of the length of side  $AB$  are  $p \pm q\sqrt{r}$ , where  $p$  and  $q$  are rational, and  $r$  is an integer not divisible by the square of a prime.  Find  $r$ .

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