You've designed a computer and implemented all the common arithmetic operators: addition, subtraction, multiplication and integer division. However, your budget was very limited, so you could only afford to place a single register in the computer. The register can store any nonnegative integer value. Since there is only one register, there is no need to identify the store location or the operands of each operation or its result. The programming language has four instructions: '+', '', '*' and '/'. Each instruction performs the corresponding operation using the value in the register as both its parameters. It then stores the result in the same register, overwriting the previous content.
A program for your computer is a sequential list of zero or more instructions. You want to show that, even with its limitations, your newly constructed computer is powerful. You will be given two ints s and t. Return the shortest program that finishes with a value of t in the register if it contained s before executing. If there is more than one possible answer, return the one that comes earliest lexicographically. If there is no program that can do the job, return ":(" (quotes for clarity) instead.
