Magician and Pigs (Hard Version)

2000ms

262144K

Description:

This is the hard version of the problem. The only difference between the two versions is the constraint on $$$n$$$ and $$$x$$$. You can make hacks only if both versions of the problem are solved.

Little09 has been interested in magic for a long time, and it's so lucky that he meets a magician! The magician will perform $$$n$$$ operations, each of them is one of the following three:

$$$1\ x$$$: Create a pig with $$$x$$$ Health Points.
$$$2\ x$$$: Reduce the Health Point of all living pigs by $$$x$$$.
$$$3$$$: Repeat all previous operations. Formally, assuming that this is the $$$i$$$-th operation in the operation sequence, perform the first $$$i-1$$$ operations (including "Repeat" operations involved) in turn.

A pig will die when its Health Point is less than or equal to $$$0$$$.

Little09 wants to know how many living pigs there are after all the operations. Please, print the answer modulo $$$998\,244\,353$$$.

Input:

The first line contains a single integer $$$n$$$ ($$$1\leq n\leq 8\cdot 10^5$$$) — the number of operations.

Each of the following $$$n$$$ lines contains an operation given in the form described in the problem statement. It's guaranteed that $$$1\leq x\leq 10^9$$$ in operations of the first two types.

Output:

Print a single integer — the number of living pigs after all the operations, modulo $$$998\,244\,353$$$.

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Sample Input:

Sample Output:

Note:

In the first example, the operations are equivalent to repeating four times: create a pig with $$$8$$$ Health Points and then reduce the Health Points of all living pigs by $$$3$$$. It is easy to find that there are two living pigs in the end with $$$2$$$ and $$$5$$$ Health Points.

Informação

Provedor Codeforces

Código CF1774F2