Arithmetically Challenged

Challenge 24 is a popular mathematics game used in many grade schools. In each game, contestants are given a card with four positive integers i_1, i_2, i_3, i_4 on it, and the first one who can use all of these numbers and any combination of the four basic arithmetic operations to get 24 wins. Each of the numbers i_1, i_2, i_3, i_4 must be used exactly once. Division can be used only if the divisor evenly divides the dividend (i.e., you can perform $6/2$ but not $6/4$). For example, if the card contains the numbers $7$, $2$, $5$, and $1$, possible solutions are (7-2)\cdot5-1 or (7+1)\cdot(5-2). Hmmm... this sounds like a source of a good programming problem.

Write a program that determines the longest consecutive sequence of integers that can be obtained by combining the four integers using arithmetic operations. For example, with $7$, $2$, $5$, and $1$, the longest consecutive sequence is from $-18$ to $26$ (yes, negative results are allowed). The "+" and "-" operators must be used as binary operators, not as unary signs.