
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
Where can we place point $C$ to construct a right triangle with hypotenuse $AB$ such that, if we draw circles centered at $A$ with radius $AC$ and at $B$ with radius $BC$, and denote the projection of $C$ onto $AB$ as $D$, the centroids of triangles $ACD$ and $BCD$ both lie in the common area of the two circles?

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