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A square is inscribed in a right triangle, as shown below. Find the area of the square. [asy] unitsize(1.5 cm); pair A, B, C, D, E, F, G; A = (0,0); C = (5,0); B = (3^2/5,3*4/5); D = extension(B, A (0,-5), A, C); G = extension(B, C (0,-5), A, C); E = extension(D, D (0,1), A, B); F = extension(G, G (0,1), B, C); draw(A--B--C--cycle); draw(D--E--F--G); draw(rightanglemark(A,B,C,5)); draw(shift((-0.2,0.1))*(A--B), Arrows(6)); draw(shift((0.1,0.2))*(B--C), Arrows(6)); label("$1$", (A B)/2 (-0.2,0.1), NW, red); label("$3$", (B C)/2 (0.1,0.2), NE, red); [/asy]
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