
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
Suppose that  $ABC$  is a triangle with  $AB = 6, BC = 12$ , and  $\angle B = 90^{\circ}$ . Point  $D$  lies on side  $BC$ , and point  $E$  is constructed on  $AC$  such that  $\angle ADE = 90^{\circ}$ . Given that  $DE = EC = \frac{a\sqrt{b}}{c}$  for positive integers  $a, b,$  and  $c$  with  $b$  squarefree and  $\gcd(a,c) = 1$ , find  $a+ b+c$ .

*Proposed by Andrew Wu*

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