Unexpected number of solutions from Solve with system of equations?

Questions
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1

I want to solve a system of three equations to get the values of three variables. It doesn’t matter whether I use Mathematica 13.1 or the Wolfram Engine 13.1, the results are the same.
Here is the code:

Solve[{(a + 1)*(b + 1)*(c + 1) == 3 &&
           (a + 2)*(b + 2)*(c + 2) == -2 &&
           (a + 3)*(b + 3)*(c + 3) == -1},
           {a, b, c}, Reals] // N

I get six solutions:

{{a -> -3.1451, b -> -1.52398, c -> 1.66908}, {a -> -3.1451, 
  b -> 1.66908, c -> -1.52398}, {a -> -1.52398, b -> -3.1451, 
  c -> 1.66908}, {a -> -1.52398, b -> 1.66908, 
  c -> -3.1451}, {a -> 1.66908, b -> -3.1451, 
  c -> -1.52398}, {a -> 1.66908, b -> -1.52398, c -> -3.1451}}

Unfortunately, none of them is correct. And why do I get six solutions?

The correct answer is: a = 1.669079088, b = -1.523976396, c = -3.145102692

Do you have any suggestion how to improve the results?

2

Your solution appears to agree with the last one provided by Solve. As for 6 solutions, fewer would make no sense unless there is multiplicity. There are three solutions and they are symmetric in the three variables. So any permutation of one solution will give another.

3

Okay, I understand. Thank you.