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문제

In this problem you are asked to help the designer of a new card game. The motive for the game is to encourage children to practice mental arithmetic. In the game, each card will present players with a range of one digit numbers and an integer value. Players must then find a way of using the basic arithmetic operations: add, subtract, multiply and modulus on the range of numbers in order to produce the given integer value. Notice that the game designer is a computer scientist and chose modulus (remainder) as the fourth arithmetic operation, rather than division as might have been expected. Your task is to write a program which finds answers for the problems on the cards.

Specifically, given a contiguous range of digits taken from 1,2,3,4,5,6,7,8 and a target integer value: find all ways of expressing the target integer value as an expression involving the one digit numbers, written in order. You may use the four binary operators: add, subtract, multiply and modulus, and brackets to control the order of evaluation. (Note: the digit 9 is never included.). For example: Given the range 1..4 and the integer 10, we note that ((((1 + 2)+3)+4) = 10. That is not the only possibility however. See sample output for the rest.

입력

The input is a sequence of problems. Each problem is described on one line of input by 3 numbers: A B C. The first two are the range of digits: 1 <= A <= B <= 8. C is the target value. Input is terminated by a line with three zeroes (not to be processed).

출력

For each problem output a line with ‘Problem #n: A..B => C’. Then output one line for every expression evaluating to the target integer. Expressions will be fully bracketed in the sense that there will be a pair of brackets enclosing each operator and its operands. Expressions will be output in order as follows. Firstly, order by bracketing pattern – working left to right digits come before brackets, so (1+(… comes before ((1+… Secondly: again working left to right, for expressions with the same bracketing pattern, + comes before * comes before – comes before %. (1+(2+… is before (1+(2*… is before (1*(2+… is before (1*(2*…

Only valid expressions with correct bracketing should be output. Expressions involving divide by zero are not considered valid.

예제 입력 1

1 4 10
2 2 2
0 0 0


예제 출력 1

Problem #1: 1..4 => 10
(1+(2+(3+4)))
(1+((2+3)+4))
(1*((2*3)+4))
((1+2)+(3+4))
((1+(2+3))+4)
((1*(2*3))+4)
(((1+2)+3)+4)
(((1*2)*3)+4)
Problem #2: 2..2 => 2
2