#P1732D2. Balance (Hard version)
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ID: 8034
Type: RemoteJudge
3000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: 9
Uploaded By:
Hydro
Tags> Balance (Hard version)
No submission language available for this problem.
Description
This is the hard version of the problem. The only difference is that in this version there are remove queries.
Initially you have a set containing one element — $0$. You need to handle $q$ queries of the following types:
- + $x$ — add the integer $x$ to the set. It is guaranteed that this integer is not contained in the set;
- - $x$ — remove the integer $x$ from the set. It is guaranteed that this integer is contained in the set;
- ? $k$ — find the $k\text{-mex}$ of the set.
In our problem, we define the $k\text{-mex}$ of a set of integers as the smallest non-negative integer $x$ that is divisible by $k$ and which is not contained in the set.
The first line contains an integer $q$ ($1 \leq q \leq 2 \cdot 10^5$) — the number of queries.
The following $q$ lines describe the queries.
An addition query of integer $x$ is given in the format + $x$ ($1 \leq x \leq 10^{18}$). It is guaranteed that $x$ is not contained in the set.
A remove query of integer $x$ is given in the format - $x$ ($1 \leq x \leq 10^{18}$). It is guaranteed that $x$ is contained in the set.
A search query of $k\text{-mex}$ is given in the format ? $k$ ($1 \leq k \leq 10^{18}$).
It is guaranteed that there is at least one query of type ?.
For each query of type ? output a single integer — the $k\text{-mex}$ of the set.
Input
The first line contains an integer $q$ ($1 \leq q \leq 2 \cdot 10^5$) — the number of queries.
The following $q$ lines describe the queries.
An addition query of integer $x$ is given in the format + $x$ ($1 \leq x \leq 10^{18}$). It is guaranteed that $x$ is not contained in the set.
A remove query of integer $x$ is given in the format - $x$ ($1 \leq x \leq 10^{18}$). It is guaranteed that $x$ is contained in the set.
A search query of $k\text{-mex}$ is given in the format ? $k$ ($1 \leq k \leq 10^{18}$).
It is guaranteed that there is at least one query of type ?.
Output
For each query of type ? output a single integer — the $k\text{-mex}$ of the set.
18
+ 1
+ 2
? 1
+ 4
? 2
+ 6
? 3
+ 7
+ 8
? 1
? 2
+ 5
? 1
+ 1000000000000000000
? 1000000000000000000
- 4
? 1
? 2
10
+ 100
? 100
+ 200
? 100
- 100
? 100
+ 50
? 50
- 50
? 50
3
6
3
3
10
3
2000000000000000000
3
4
200
300
100
100
50
Note
In the first example:
After the first and second queries, the set will contain elements $\{0, 1, 2\}$. The smallest non-negative number that is divisible by $1$ and is not in the set is $3$.
After the fourth query, the set will contain the elements $\{0, 1, 2, 4\}$. The smallest non-negative number that is divisible by $2$ and is not in the set is $6$.
In the second example:
- Initially, the set contains only the element $\{0\}$.
- After adding an integer $100$ the set contains elements $\{0, 100\}$.
- $100\text{-mex}$ of the set is $200$.
- After adding an integer $200$ the set contains elements $\{0, 100, 200\}$.
- $100\text{-mex}$ of the set $300$.
- After removing an integer $100$ the set contains elements $\{0, 200\}$.
- $100\text{-mex}$ of the set is $100$.
- After adding an integer $50$ the set contains elements $\{0, 50, 200\}$.
- $50\text{-mex}$ of the set is $100$.
- After removing an integer $50$ the set contains elements $\{0, 200\}$.
- $100\text{-mex}$ of the set is $50$.