12. If:
(1) $a, b, c, d$ all belong to $\{1,2,3,4\}$;
(2) $a \neq b, b \neq c, c \neq d, d \neq a$;
(3) $a$ is the smallest value among $a, b, c, d$.
Then, the number of different four-digit numbers $\overline{a b c d}$ that can be formed is $\qquad$.
When $a, b, c, d$ contain 2 different digits $x<y$, it can only be in the form $\overline{x y x y}$, in which case there are $C_{4}^{2}=6$ different four-digit numbers;
When $a, b, c, d$ contain 3 different digits $x, y, z$ (with $x$ being the smallest), there can be 4 forms: $\overline{x y x z}, \overline{x z x y}$, $\overline{x y z y}, \overline{x z y z}$, in which case there are $C_{4}^{3} \cdot 4=16$ different four-digit numbers;
When $a, b, c, d$ contain 4 different digits, apart from the first digit 1, the latter three can be arranged in any order. In this case, there are $A_{3}^{2}=6$ different four-digit numbers.
Therefore, the total number of different four-digit numbers $\overline{a b c d}$ that can be formed is $6+16+6=28$.