Cat and Claire are having a conversation about Cat's favorite triangle.
Cat says, "My favorite triangle is a right triangle with integer side lengths, all of which are less than or equal to ."
Claire says, "I don't know your favorite triangle yet, but I do know that some triangles with the properties you mentioned have the property such that the numerical value of their perimeter is two-thirds of the numerical value of their area."
Cat says, "That's cool, but my favorite triangle doesn't have that property. Also, if I told you what any of the side lengths of my favorite triangle were, you'd immediately know all its lengths."
Claire says, "Now I know all the side lengths of your favorite triangle!"
What is the area of Cat's favorite triangle?