no computers - French alphametic (misspelled)

The following puzzle has a spelling mistake: DOUCE should be DOUZE.
The correct version is: French alphametic (corrected)

(Surprisingly the faulty version allows feasible solutions.)

Every letter stands for a digit in base-10 representation, different letters stand for different digits:

         UN 
         UN
       DEUX
   +  DOUCE
  ------------
      SEIZE

Which digit does each letter represent?
(Please present the full analysis how these digits can be determined.)

Answer

sorry for the format I don't know how to put spoiler to multiple lines of formula. I am just putting spoiler to the answer below.

remaining numbers : 0-1-2-3-4-5-6-7-8-9

             UN
             UN
           DEUX
        + DOUCE
        -------

          SEIZE

Giving 0 to E , and since we know S = D + 1 I random them as 8 - 9. I will try giving 2 to O . This is what we have:

E=0 , D=8 , S=9, O=2, remaining numbers : 1-3-4-5-6-7

              UN
              UN
            80UX
        +  82UC0
         -------
           90IZ0

Now we know that :

N+N+X = 10 or 20

U+U+U+C + B = Z where B = 1 or 2

U + 0 = I but we know they can't be the same so there is a remainder which means:

U + W = I where W = 1 or 2

        N N X possibilities:
        a1: 7+7+6
        a2: 3+3+4
        a3: 2+2+6

To determine whether if W or B can be 1 or 2 I pick a2 to give big numbers to net stage and I get;

E=0 , D=8 , S=9, O=2, N=3, X=4, remaining numbers : 1-5-6-7

             U3
             U3
           80U4
        + 82UC0
        -------
          90IZ0

U+U+U+C + 1 = Z

U + W = I where W = 1 or 2

U can be 5 or 6, and I can 6 or 7 because of remaining numbers.

UUUC possibilities:

6661 +1 = 20 x Z can't be 0
6665 +1 = 24 x Z can't be 4
5557 +1 = 23 x Z can't be 3
5551 +1 = 17 so let's give our values and rewrite:

ANSWER:

E=0 , D=8 , S=9, O=2, N=3, X=4, U=5, C=1, Z=7, I=6

             53
             53
           8054
        + 82510
        -------
          90670

Is my solution but I am sure there are more possibilities and this is not a unique solution because of many possibilities.