Description:
The only difference between this problem and the easy version is the constraints on $$$t$$$ and $$$n$$$.
You are given an array $$$a$$$, consisting of $$$n$$$ distinct integers $$$a_1, a_2, \ldots, a_n$$$.
Define the beauty of an array $$$p_1, p_2, \ldots p_k$$$ as the minimum amount of time needed to sort this array using an arbitrary number of range-sort operations. In each range-sort operation, you will do the following:
- Choose two integers $$$l$$$ and $$$r$$$ ($$$1 \le l < r \le k$$$).
- Sort the subarray $$$p_l, p_{l + 1}, \ldots, p_r$$$ in $$$r - l$$$ seconds.
Please calculate the sum of beauty over all subarrays of array $$$a$$$.
A subarray of an array is defined as a sequence of consecutive elements of the array.
Input Format:
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.
The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 3 \cdot 10^5$$$) — the length of the array $$$a$$$.
The second line of each test case consists of $$$n$$$ integers $$$a_1,a_2,\ldots, a_n$$$ ($$$1\le a_i\le 10^9$$$). It is guaranteed that all elements of $$$a$$$ are pairwise distinct.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$3 \cdot 10^5$$$.
Output Format:
For each test case, output the sum of beauty over all subarrays of array $$$a$$$.
Note:
In the first test case:
- The subarray $$$[6]$$$ is already sorted, so its beauty is $$$0$$$.
- The subarray $$$[4]$$$ is already sorted, so its beauty is $$$0$$$.
- You can sort the subarray $$$[6, 4]$$$ in one operation by choosing $$$l = 1$$$ and $$$r = 2$$$. Its beauty is equal to $$$1$$$.
In the second test case:
- The subarray $$$[3]$$$ is already sorted, so its beauty is $$$0$$$.
- The subarray $$$[10]$$$ is already sorted, so its beauty is $$$0$$$.
- The subarray $$$[6]$$$ is already sorted, so its beauty is $$$0$$$.
- The subarray $$$[3, 10]$$$ is already sorted, so its beauty is $$$0$$$.
- You can sort the subarray $$$[10, 6]$$$ in one operation by choosing $$$l = 1$$$ and $$$r = 2$$$. Its beauty is equal to $$$2 - 1 = 1$$$.
- You can sort the subarray $$$[3, 10, 6]$$$ in one operation by choosing $$$l = 2$$$ and $$$r = 3$$$. Its beauty is equal to $$$3 - 2 = 1$$$.