
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
A circle is circumscribed about a right triangle with sides 15, 36, and 39. An altitude is drawn from the right angle vertex to the hypotenuse, dividing one of the non-triangular area regions into two parts, say $A$ (smaller) and $B$ (larger). Let $C$ denote the largest non-triangular region. If area $A$ is the region adjacent to the vertices of sides $15$ and $36$, find the relation between $A, B,$ and $C$.
A) $A + B = C$
B) $2A = C - 270$
C) $A + B + 270 = C$
D) $A^2 + B^2 = C^2$
E) $\frac{1}{A^2} + \frac{1}{B^2} = \frac{1}{C^2}$

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