<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let  $T=TNFTPP$ .  Points  $A$  and  $B$  lie on a circle centered at  $O$  such that  $\angle AOB$  is right.  Points  $C$  and  $D$  lie on radii  $OA$  and  $OB$  respectively such that  $AC = T-3$ ,  $CD = 5$ , and  $BD = 6$ .  Determine the area of quadrilateral  $ACDB$ .

[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=10$</details>. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
44
Let  $T=TNFTPP$ .  Points  $A$  and  $B$  lie on a circle centered at  $O$  such that  $\angle AOB$  is right.  Points  $C$  and  $D$  lie on radii  $OA$  and  $OB$  respectively such that  $AC = T-3$ ,  $CD = 5$ , and  $BD = 6$ .  Determine the area of quadrilateral  $ACDB$ .

[asy]
draw(circle((0,0),10));
draw((0,10)--(0,0)--(10,0)--(0,10));
draw((0,3)--(4,0));
label("O",(0,0),SW);
label("C",(0,3),W);
label("A",(0,10),N);
label("D",(4,0),S);
label("B",(10,0),E);
[/asy]

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, <details><summary>here's the value of T</summary>$T=10$</details>.
test