
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 points  $A$  and  $B$  lie on circle  $\Omega$ , and suppose that points  $C$  and  $D$  are the trisection points of major arc  $AB$ , with  $C$  closer to  $B$  than  $A$ . Let  $E$  be the intersection of line  $AB$  with the line tangent to  $\Omega$  at  $C$ . Suppose that  $DC = 8$  and  $DB = 11$ . If  $DE = a\sqrt{b}$  for integers  $a$  and  $b$  with  $b$  squarefree, find  $a + b$ .

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