Arithmetic expressions

By using each of the digits from the set, ${1, 2, 3, 4}$, exactly once, and making use of the four arithmetic operations (+, −, *, /) and brackets/parentheses, it is possible to form different positive integer targets.

For example,

$$ \begin{aligned} &8 = (4 * (1 + 3)) / 2\ &14 = 4 * (3 + 1 / 2)\ &19 = 4 * (2 + 3) − 1\ &36 = 3 * 4 * (2 + 1)\ \end{aligned} $$

Note that concatenations of the digits, like $12 + 34$, are not allowed.

Using the set, ${1, 2, 3, 4}$, it is possible to obtain thirty-one different target numbers of which $36$ is the maximum, and each of the numbers $1$ to $28$ can be obtained before encountering the first non-expressible number.

Find the set of four distinct digits, $a < b < c < d$, for which the longest set of consecutive positive integers, $1$ to $n$, can be obtained, giving your answer as a string: abcd.