Robot Manhattan Path Sum

A robot is placed on a 2D coordinate system at the origin $(0,0)$. It receives a sequence of commands represented by the characters S, J, I, and Z. These instructions indicate movements in the following way:

• S : move from $(x,y)$ to $(x, y+1)$,
• J : move from $(x,y)$ to $(x, y-1)$,
• I : move from $(x,y)$ to $(x+1, y)$, and
• Z : move from $(x,y)$ to $(x-1, y)$.

There are $n$ fixed control points on the track. Each control point measures the Manhattan distance to the robot. The Manhattan distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

After executing each command, compute and output the sum of Manhattan distances from the robot's current position to all control points.

Note: The input starts with two integers $n$ and $m$, where $n$ is the number of control points and $m$ is the number of commands. Then follow $n$ lines each containing two integers representing the coordinates of a control point. The final line contains a string of $m$ characters (each being one of S, J, I, or Z) representing the sequence of commands.

inputFormat

The first line contains two integers $n$ and $m$.

The next $n$ lines each contain two integers $x$ and $y$, representing the coordinates of a control point.

The last line contains a string of length $m$ consisting only of the characters S, J, I, and Z indicating the robot's commands.

outputFormat

Output $m$ lines. The $i$-th line should contain the sum of Manhattan distances from the robot’s current position (after executing the first $i$ commands) to all control points.

sample

2 3
1 0
0 1
SIJ

2
2
2