MEX Table
Description
One day, the schoolboy Mark misbehaved, so the teacher Sasha called him to the whiteboard.
Sasha gave Mark a table with $n$ rows and $m$ columns. His task is to arrange the numbers $0, 1, \ldots, n \cdot m - 1$ in the table (each number must be used exactly once) in such a way as to maximize the sum of MEX$^{\text{∗}}$ across all rows and columns. More formally, he needs to maximize $$\sum\limits_{i = 1}^{n} \operatorname{mex}(\{a_{i,1}, a_{i,2}, \ldots, a_{i,m}\}) + \sum\limits_{j = 1}^{m} \operatorname{mex}(\{a_{1,j}, a_{2,j}, \ldots, a_{n,j}\}),$$ where $a_{i,j}$ is the number in the $i$-th row and $j$-th column.
Sasha is not interested in how Mark arranges the numbers, so he only asks him to state one number — the maximum sum of MEX across all rows and columns that can be achieved.
$^{\text{∗}}$The minimum excluded (MEX) of a collection of integers $c_1, c_2, \ldots, c_k$ is defined as the smallest non-negative integer $x$ which does not occur in the collection $c$.
For example:
- $\operatorname{mex}([2,2,1])= 0$, since $0$ does not belong to the array.
- $\operatorname{mex}([3,1,0,1]) = 2$, since $0$ and $1$ belong to the array, but $2$ does not.
- $\operatorname{mex}([0,3,1,2]) = 4$, since $0$, $1$, $2$, and $3$ belong to the array, but $4$ does not.
Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 1000$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $m$ ($1 \le n, m \le 10^9$) — the number of rows and columns in the table, respectively.
For each test case, output the maximum possible sum of $\operatorname{mex}$ across all rows and columns.
Input
Each test contains multiple test cases. The first line contains a single integer $t$ ($1 \le t \le 1000$) — the number of test cases. The description of the test cases follows.
The first line of each test case contains two integers $n$ and $m$ ($1 \le n, m \le 10^9$) — the number of rows and columns in the table, respectively.
Output
For each test case, output the maximum possible sum of $\operatorname{mex}$ across all rows and columns.
3
1 1
2 2
3 5
2
3
6
Note
In the first test case, the only element is $0$, and the sum of the $\operatorname{mex}$ of the numbers in the first row and the $\operatorname{mex}$ of the numbers in the first column is $\operatorname{mex}(\{0\}) + \operatorname{mex}(\{0\}) = 1 + 1 = 2$.
In the second test case, the optimal table may look as follows:
| $3$ | $0$ |
| $2$ | $1$ |
Then $\sum\limits_{i = 1}^{n} \operatorname{mex}(\{a_{i,1}, a_{i,2}, \ldots, a_{i,m}\}) + \sum\limits_{j = 1}^{m} \operatorname{mex}(\{a_{1,j}, a_{2,j}, \ldots, a_{n,j}\}) = \operatorname{mex}(\{3, 0\}) + \operatorname{mex}(\{2, 1\})$ $+ \operatorname{mex}(\{3, 2\}) + \operatorname{mex}(\{0, 1\}) = 1 + 0 + 0 + 2 = 3$.