
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 a triangle with  $AB = 5$ ,  $AC = 8$ , and  $BC = 7$ . Let  $D$  be on side  $AC$  such that  $AD = 5$  and  $CD = 3$ . Let  $I$  be the incenter of triangle  $ABC$  and  $E$  be the intersection of the perpendicular bisectors of  $\overline{ID}$  and  $\overline{BC}$ .  Suppose  $DE = \frac{a\sqrt{b}}{c}$  where  $a$  and  $c$  are relatively prime positive integers, and  $b$  is a positive integer not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Ray Li*

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