
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]


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In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths  $a$  and  $b$ . If the right triangle has area  $T$  and is inscribed in a circle of area  $C$ , find  $ab$  in terms of  $T$  and  $C$ .

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