Background

Hi, I'm trying to solve a programming problem and I'm stuck on the following problem:

Assume you have multiple lists of numbers. All are sorted in decreasing order. You now have to take exactly one number from each list to make the biggest possible sum.

So far so easy, to solve this you could just take the first number of each list and you're done.

But now, I need the second-largest sum while still using exactly one number from each list. To achieve this, I would take the first element in each list but for the list which has the least difference between the first and second number the second number will be used.

This is still pretty doable.

The Problem

But I need an Iterator over every possible sum using exactly one number of each list sorted in decreasing order.
For performance reasons, it isn't possible to just compute every sum and then sort it. The algorithm must already provide the sums in decreasing order. If there are multiple combinations for a sum then the sum must be returned multiple times.

Additional Requirements

The Iterator should be lazy (only calculate the next sum when required).
The Lists are already lazy, which means you should require as few values as possible to calculate the fitting sum.

Example

For the Lists:

List 1: [5, 2, 1]
List 2: [10, 2]
List 3: [6, 1]

The Iterator then should return:

[5, 10, 6] = 21
[2, 10, 6] = 18
[1, 10, 6] = 17
[5, 10, 1] = 16
[5,  2, 6] = 13
[2, 10, 1] = 13
[1, 10, 1] = 12
[2,  2, 6] = 10
[1,  2, 6] = 9
[5,  2, 1] = 8
[2,  2, 1] = 5
[1,  2, 1] = 4

Comment

I don't need code as an answer to my question (you're still welcome to provide it if it helps to explain). What I'm looking for are ideas to solve this, or solutions that I can implement myself.

Thanks in advance!