Eccoci.

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

For example,

\begin{align} 8 &= (4 \times (1 + 3)) / 2\\ 14 &= 4 \times (3 + 1 / 2)\\ 19 &= 4 \times (2 + 3) - 1\\ 36 &= 3 \times 4 \times (2 + 1) \end{align}

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