P16753 [GKS 2020 #B] Robot Path Decoding

Your country's space agency has just landed a rover on a new planet. The planet's surface can be thought of as a grid of squares containing $10^9$ columns (numbered starting from $1$ from west to east) and $10^9$ rows (numbered starting from $1$ from north to south). Let $(w, h)$ denote the square in the $w$-th column and the $h$-th row. The rover begins on the square $(1, 1)$. The rover can be maneuvered around on the surface of the planet by sending it a program, which contains a string of characters representing movements in the four cardinal directions. The robot executes each character of the string in order. The rover moves according to the following rules: * N: Move one unit north. * S: Move one unit south. * E: Move one unit east. * W: Move one unit west. There is also a special instruction $\text{X}(\text{Y})$, where $\text{X}$ is a number between $2$ and $9$ (inclusive) and $\text{Y}$ is a non-empty subprogram. This denotes that the robot should repeat the subprogram $\text{Y}$ a total of $\text{X}$ times. For example: * $2(\text{NWE})$ is equivalent to NWENWE. * $3(\text{S } 2(\text{E}))$ is equivalent to SEESEESEE. * EEEE$4(\text{N})$ $2(\text{SS})$ is equivalent to EEEENNNNSSSS. Since the planet is a spheroid, the first and last columns are adjacent, so moving east from column $10^9$ will move the rover to column $1$ and moving south from row $10^9$ will move the rover to row $1$. Similarly, moving west from column $1$ will move the rover to column $10^9$ and moving north from row $1$ will move the rover to row $10^9$. Given a program that the robot will execute, determine the final position of the robot after it has finished all its movements.

The first line of the input gives the number of test cases, $T$. $T$ lines follow. Each line contains a single string: the program sent to the rover.

For each test case, output one line containing Case #x: w h, where x is the test case number (starting from 1) and w h is the final square $(w, h)$ the rover finishes in.

In Sample Case #1, the rover moves three units south, then three units east. In Sample Case #2, the rover moves one unit north. Since the planet is a torus, this moves it into row $10^9$. In Sample Case #3, the program given to the rover is equivalent to NSSSNEEN. In Sample Case #4, the program given to the rover is equivalent to NWNWNWWEEEEWWEEEEWNWNWNWWEEEEWWEEEEW ### Limits $1 \le T \le 100$. The string represents a valid program. The length of each program is between $1$ and $2000$ characters inclusive. **Test Set 1** The total number of moves the robot will make in a single test case is at most $10^4$. **Test Set 2** No additional constraints.