I have written down another inequality that is also true, but I'm not showing it to you. This inequality puts the following values in order from greatest to smallest:

A, A+C, B, A+D, C, B+C, D, B+D, A+B, C+D

(this is obviously not the order, as A+C can't be less than A or less than C)

I showed the two inequalities to my friend, and he was able to minimize A, B, C, and D all at once. I told him that he had just guessed correctly which numbers were on my mind.

Based only on this information, what is the highest possible sum of the four numbers on my mind?