Twenty Four, Again

Yes, we know $\ldots $
we’ve used Challenge $24$
before for contest problems. In case you’ve never heard of
Challenge $24$ (or have a
very short memory) the object of the game is to take
$4$ given numbers (the
*base values*) and determine if there is a way to
produce the value $24$
from them using the four basic arithmetic operations (and
parentheses if needed). For example, given the four base values
`3 5 5 2`, you can produce $24$ in many ways. Two of them are:
`5*5-3+2` and `(3+5)*(5-2)`. Recall that
multiplication and division have precedence over addition and
subtraction, and that equal-precedence operators are evaluated
left-to-right.

This is all very familiar to most of you, but what you
probably don’t know is that you can *grade* the quality
of the expressions used to produce $24$. In fact, we’re sure you don’t
know this since we’ve just made it up. Here’s how it works: A
perfect grade for an expression is $0$. Each use of parentheses adds one
point to the grade. Furthermore, each inversion (that is, a
swap of two adjacent values) of the original ordering of the
four base values adds two points. The first expression above
has a grade of $4$, since
two inversions are used to move the $3$ to the third position. The second
expression has a better grade of $2$ since it uses no inversions but
two sets of parentheses. As a further example, the initial set
of four base values `3 6 2 3` could produce an
expression of grade $3$ —
namely `(3+6+3)*2` — but it also has a perfect grade
$0$ expression — namely
`3*6+2*3`. Needless to say, the lower the grade the
“better” the expression.

Two additional rules we’ll use: $1$) you cannot use unary minus in any
expression, so you can’t take the base values `3 5 5 2`
and produce the expression `-3+5*5+2`, and $2$) division can only be used if the
result is an integer, so you can’t take the base values `2 3
4 9` and produce the expression `2/3*4*9`.

Given a sequence of base values, determine the lowest graded
expression resulting in the value $24$. And by the way, the initial set
of base values `3 5 5 2` has a grade $1$ expression — can you find it?

Input consists of a single line containing $4$ base values. All base values are between $1$ and $100$, inclusive.

Display the lowest grade possible using the sequence of base
values. If it is not possible to produce $24$, display `impossible`.

Sample Input 1 | Sample Output 1 |
---|---|

3 5 5 2 |
1 |

Sample Input 2 | Sample Output 2 |
---|---|

1 1 1 1 |
impossible |