Determine all possible sextuplets (A, B, C, D, E, F) of positive integers, with A ≤ B ≤ C, and, D ≤ E ≤ F and, A ≤ D, that satisfy both the equations: A+B+C = D*E*F and, A*B*C = D+E+F.

Prove that these are the only sextuplets that exist.

A computer program kept giving me the following six solutions for (A, B, C; D, E, F):

(1, 1, 6; 2, 2, 2)

(1, 1, 7; 1, 3, 3)

(1, 1, 8; 1, 2, 5)

(1, 2, 3; 1, 2, 3)

(1, 2, 5; 1, 1, 8)

(1, 3, 3; 1, 1, 7)

However the last two are just repeats of the 2nd and 3rd ones, so there are really four unique solutions. After running the program for possible values <=25 with the same results, I'm pretty certain they must be the only solutions. I have no idea how to prove this analytically, but my gut feeling is that if any variable is 10 or more, it makes the equations too lop-sided to be in agreement.