\(ABC\) is a right triangle with integer side lengths. The area of \(\triangle ABC\) is equal to its perimeter.

What is the sum of all possible areas of \(\triangle ABC\)?

This problem is a part of the set "I Don't Have a Good Name For This Yet". See the rest of the problems here. And when I say I don't have a good name for this yet, I mean it. If you like problems like these and have a cool name for this set, feel free to comment here.

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