The Industrial Computer Processor Company offers very fast, special purpose processing units tailored to customer needs. Processors of the \textbf{a-C-m} family (such as the \textbf{1-C-2} and the \textbf{5-C-3}) have an instruction set with only two different operations: \textbf{A add a} \textbf{M multiply by m} The processor receives an integer, executes a sequence of \textbf{A} and \textbf{M} operations (the program) that modifies the input, and outputs the result. For example, the \textbf{1-C-2} processor executing the program \textbf{AAAM} with the input \textbf{2} yields the output \textbf{10} (the computation is \textbf{2} → \textbf{3} → \textbf{4} → \textbf{5} → \textbf{10}), while the \textbf{5-C-3} processor yields \textbf{51} with the same program and input (\textbf{2} → \textbf{7} → \textbf{12} → \textbf{17} → \textbf{51}). You are an \textbf{a-C-m} programmer assigned to a top secret project. This means that you have not been told the precise computation your program should perform. But you are given particular values \textbf{p}, \textbf{q}, \textbf{r}, and \textbf{s} and the following conditions: \begin{enumerate} \item The input is guaranteed to be a number between \textbf{p} and \textbf{q}. \item The output must be some number between \textbf{r} and \textbf{s}. \end{enumerate} Given an \textbf{a-C-m} processor and the numbers \textbf{p}, \textbf{q}, \textbf{r}, and \textbf{s}, your job is to construct the shortest \textbf{a-C-m} program which, for every input \textbf{x} such that \textbf{p} ≤ \textbf{x} ≤ \textbf{q}, yields some output such that \textbf{r} ≤ \textbf{y} ≤ \textbf{s}. If there is more than one program of minimum length, choose the one that come first lexicographically, treating each program as a string of \textbf{As} and \textbf{Ms}. \InputFile The input contains several test cases. Each test case is given by a line with the six integers \textbf{a}, \textbf{m}, \textbf{p}, \textbf{q}, \textbf{r}, and \textbf{s} as described above (\textbf{1} ≤ \textbf{a}, \textbf{m}, \textbf{p}, \textbf{q}, \textbf{r}, \textbf{s} ≤ \textbf{10^9}, \textbf{p} ≤ \textbf{q} and \textbf{r} ≤ \textbf{s}). The last test case is followed by a line with six zeros. \OutputFile For each test case, display its case number followed by the best program as described above. Display the word "\textbf{empty}" if the best program uses no operations. Display the word "\textbf{impossible}" if there is no program meeting the specifications. Display the program as a sequence of space-separated strings, alternating between strings of the form "\textbf{nA}" and strings of the form "\textbf{nM}", where \textbf{n} > \textbf{0}. Strings of the former type indicate \textbf{n} consecutive \textbf{A} operations, and strings of the latter type indicate consecutive \textbf{M} operations. Follow the format of the sample output.