You are going to a stone skipping contest. The lake can be represented as a number line, with the shore being all negative integers, and the lake occupying all the nonnegative integers.

You may stand at any integer position on the shore and throw your stone with any integer power. If you throw with power $$$p$$$, the stone will move $$$p$$$ meters before bouncing. When the stone bounces, if it has power $$$p$$$, it will bounce again with power $$$\lfloor p/2 \rfloor$$$, and if the stone has power $$$0$$$, it will sink. Contest regulations state the stone must sink at position $$$N$$$ $$$(1 \le N \le 10^{18})$$$, and the score of the throw is the sum of $$$a_x$$$ for each position $$$x$$$ the stone bounces, for an zero-index array $$$a$$$ of length $$$N$$$.

The array $$$a$$$ initially begins at all $$$0$$$s, but there are $$$Q$$$ $$$(1 \le Q \le 10^5)$$$ modifications which change $$$a_x = y$$$ where $$$0 \le x \lt N$$$ and $$$0 \le y \le 10^{15}$$$. Compute the maximum score after each modification.