A Quartet of Integers

I have a quartet of integers with which I’ll present a little number problem for you to solve. Here are some facts to work with:

  1. Twice the first integer, minus the second, equals 7.
  2. Twice the second integer, minus the third, equals 4.
  3. Twice the third integer, minus the fourth, equals 3.

Now there are an infinite number of quartets of integers that share those three facts, so I will now add an additional condition:

When you subtract the first integer of my particular group from twice the fourth integer, you get the minimum positive integral value for this operation of all possible cases.

What is the sum of my four integers?

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