
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  $A, B, C, D$  be points, in order, on a straight line such that  $AB=BC=CD$ . Let  $E$  be a point closer to  $B$  than  $D$  such that  $BE=EC=CD$  and let  $F$  be the midpoint of  $DE$ . Let  $AF$  intersect  $EC$  at  $G$  and let  $BF$  intersect  $EC$  at  $H$ . If  $[BHC]+[GHF]=1$ , then  $AD^2 = \frac{a\sqrt{b}}{c}$  where  $a,b,$  and  $c$  are positive integers,  $a$  and  $c$  are relatively prime, and  $b$  is not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by **AOPS12142015***

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