
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, let the  $A$ -altitude meet  $BC$  at  $D$ , let the  $B$ -altitude meet  $AC$  at  $E$ , and let  $T\neq A$  be the point on the circumcircle of  $ABC$  such that  $AT || BC$ . Given that  $D,E,T$  are collinear, if  $BD=3$  and  $AD=4$ , then the area of  $ABC$  can be written as  $a+\sqrt{b}$ , where  $a$  and  $b$  are positive integers. What is  $a+b$ ?


*2021 CCA Math Bonanza Individual Round #12*

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