Balls and Pockets

There is a strip with an infinite number of cells. Cells are numbered starting with $$$0$$$. Initially the cell $$$i$$$ contains a ball with the number $$$i$$$.

There are $$$n$$$ pockets located at cells $$$a_1, \ldots, a_n$$$. Each cell contains at most one pocket.

Filtering is the following sequence of operations:

• All pockets at cells $$$a_1, \ldots, a_n$$$ open simultaneously, which makes balls currently located at those cells disappear. After the balls disappear, the pockets close again.
• For each cell $$$i$$$ from $$$0$$$ to $$$\infty$$$, if the cell $$$i$$$ contains a ball, we move that ball to the free cell $$$j$$$ with the lowest number. If there is no free cell $$$j < i$$$, the ball stays at the cell $$$i$$$.

Note that after each filtering operation each cell will still contain exactly one ball.

For example, let the cells $$$1$$$, $$$3$$$ and $$$4$$$ contain pockets. The initial configuration of balls is shown below (underscores display the cells with pockets):

After opening and closing the pockets, balls 1, 3 and 4 disappear:

After moving all the balls to the left, the configuration looks like this:

Another filtering repetition results in the following:

You have to answer $$$m$$$ questions. The $$$i$$$-th of these questions is "what is the number of the ball located at the cell $$$x_i$$$ after $$$k_i$$$ repetitions of the filtering operation?"