
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  $T=TNFTPP$ .  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>.

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