
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  $AB = 5$ ,  $BC = 7$ , and  $CA = 8$ , and let  $D$  be a point on arc  $\widehat{BC}$  of its circumcircle  $\Omega$ . Suppose that the angle bisectors of  $\angle ADB$  and  $\angle ADC$  meet  $AB$  and  $AC$  at  $E$  and  $F$ , respectively, and that  $EF$  and  $BC$  meet at  $G$ . Line  $GD$  meets  $\Omega$  at  $T$ . If the maximum possible value of  $AT^2$  can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $\gcd (a,b) = 1$ , find  $a + b$ .

*Proposed by Andrew Wu*

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