Divan and bitwise operations

1000ms

262144K

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 $$$10^9 + 7$$$.

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 $$$10^9 + 7$$$.

Sample Input:

Sample Output:

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 $$$[5, \, 6, \, 7, \, 0, \, 2]$$$.

Informação

Provedor Codeforces

Código CF1614C