#P1614C. Divan and bitwise operations
Divan and bitwise operations
Description
Once Divan analyzed a sequence $a_1, a_2, \ldots, a_n$ consisting of $n$ non-negative integers as follows. He considered each non-empty subsequence of the sequence $a$, computed the bitwise XOR of its elements and added up all the XORs, obtaining the coziness of the sequence $a$.
A sequence $c$ is a subsequence of a sequence $d$ if $c$ can be obtained from $d$ by deletion of several (possibly, zero or all) elements. For example, $[1, \, 2, \, 3, \, 4]$, $[2, \, 4]$, and $[2]$ are subsequences of $[1, \, 2, \, 3, \, 4]$, but $[4, \, 3]$ and $[0]$ are not.
Divan was very proud of his analysis, but now he lost the sequence $a$, and also the coziness value! However, Divan remembers the value of bitwise OR on $m$ contiguous subsegments of the sequence $a$. It turns out that each element of the original sequence is contained in at least one of these $m$ segments.
Divan asks you to help find the coziness of the sequence $a$ using the information he remembers. If several coziness values are possible, print any.
As the result can be very large, print the value modulo $10^9 + 7$.
The first line contains one integer number $t$ ($1 \le t \le 10^3$) — the number of test cases.
The first line of each test case contains two integer numbers $n$ and $m$ ($1 \le n, m \le 2 \cdot 10^5$) — the length of the sequence and the number of contiguous segments whose bitwise OR values Divan remembers, respectively.
The following $m$ lines describe the segments, one per line.
Each segment is described with three integers $l$, $r$, and $x$ ($1 \le l \le r \le n$, $0 \le x \le 2^{30} - 1$) — the first and last elements of the segment and the bitwise OR of $a_l, a_{l + 1}, \ldots, a_r$, respectively.
It is guaranteed that each element of the sequence is contained in at least one of the segments. It is guaranteed that there exists a sequence that satisfies all constraints.
It is guaranteed that the sum of $n$ and the sum of $m$ over all test cases do not exceed $2 \cdot 10^5$.
For each test case print the coziness any suitable sequence $a$ modulo $10^9 + 7$.
Input
The first line contains one integer number $t$ ($1 \le t \le 10^3$) — the number of test cases.
The first line of each test case contains two integer numbers $n$ and $m$ ($1 \le n, m \le 2 \cdot 10^5$) — the length of the sequence and the number of contiguous segments whose bitwise OR values Divan remembers, respectively.
The following $m$ lines describe the segments, one per line.
Each segment is described with three integers $l$, $r$, and $x$ ($1 \le l \le r \le n$, $0 \le x \le 2^{30} - 1$) — the first and last elements of the segment and the bitwise OR of $a_l, a_{l + 1}, \ldots, a_r$, respectively.
It is guaranteed that each element of the sequence is contained in at least one of the segments. It is guaranteed that there exists a sequence that satisfies all constraints.
It is guaranteed that the sum of $n$ and the sum of $m$ over all test cases do not exceed $2 \cdot 10^5$.
Output
For each test case print the coziness any suitable sequence $a$ modulo $10^9 + 7$.
Samples
3
2 1
1 2 2
3 2
1 3 5
2 3 5
5 4
1 2 7
3 3 7
4 4 0
4 5 2
4
20
112
Note
In first example, one of the sequences that fits the constraints is $[0, 2]$. Consider all its non-empty subsequences:
- $[0]$: the bitwise XOR of this subsequence is $0$;
- $[2]$: the bitwise XOR of this subsequence is $2$;
- $[0, 2]$: the bitwise XOR of this subsequence is $2$.
The sum of all results is $4$, so it is the answer.
In second example, one of the sequences that fits the constraints is $[0, \, 5, \, 5]$.
In third example, one of the sequences that fits the constraints is $[5, \, 6, \, 7, \, 0, \, 2]$.