You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 2.0 seconds
Memory limit: 256.0 MB

# Problem
This is the hard version of the problem. The only difference between the two versions is the constraint on $$$t$$$ and $$$n$$$. You can make hacks only if both versions of the problem are solved.

For a binary$$$^\dagger$$$ pattern $$$p$$$ and a binary string $$$q$$$, both of length $$$m$$$, $$$q$$$ is called $$$p$$$-good if for every $$$i$$$ ($$$1 \leq i \leq m$$$), there exist indices $$$l$$$ and $$$r$$$ such that:

- $$$1 \leq l \leq i \leq r \leq m$$$, and
- $$$p_i$$$ is a mode$$$^\ddagger$$$ of the string $$$q_l q_{l+1} \ldots q_{r}$$$.

For a pattern $$$p$$$, let $$$f(p)$$$ be the minimum possible number of $$$\mathtt{1}$$$s in a $$$p$$$-good binary string (of the same length as the pattern).

You are given a binary string $$$s$$$ of size $$$n$$$. Find $$$$$$\sum_{i=1}^{n} \sum_{j=i}^{n} f(s_i s_{i+1} \ldots s_j).$$$$$$ In other words, you need to sum the values of $$$f$$$ over all $$$\frac{n(n+1)}{2}$$$ substrings of $$$s$$$.

$$$^\dagger$$$ A binary pattern is a string that only consists of characters $$$\mathtt{0}$$$ and $$$\mathtt{1}$$$.

$$$^\ddagger$$$ Character $$$c$$$ is a mode of string $$$t$$$ of length $$$m$$$ if the number of occurrences of $$$c$$$ in $$$t$$$ is at least $$$\lceil \frac{m}{2} \rceil$$$. For example, $$$\mathtt{0}$$$ is a mode of $$$\mathtt{010}$$$, $$$\mathtt{1}$$$ is not a mode of $$$\mathtt{010}$$$, and both $$$\mathtt{0}$$$ and $$$\mathtt{1}$$$ are modes of $$$\mathtt{011010}$$$.

## Input Format
Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^5$$$) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer $$$n$$$ ($$$1 \le n \le 10^6$$$) — the length of the binary string $$$s$$$.

The second line of each test case contains a binary string $$$s$$$ of length $$$n$$$ consisting of only characters $$$\mathtt{0}$$$ and $$$\mathtt{1}$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^6$$$.

## Output Format
For each test case, output the sum of values of $$$f$$$ over all substrings of $$$s$$$.

## Examples
```input
4
1
1
2
10
5
00000
20
11110110000000111111
```
```output
1
2
0
346
```

## Note
In the first test case, the only $$$\mathtt{1}$$$-good string is $$$\mathtt{1}$$$. Thus, $$$f(\mathtt{1})=1$$$.

In the second test case, $$$f(\mathtt{10})=1$$$ because $$$\mathtt{01}$$$ is $$$\mathtt{10}$$$-good, and $$$\mathtt{00}$$$ is not $$$\mathtt{10}$$$-good. Thus, the answer is $$$f(\mathtt{1})+f(\mathtt{10})+f(\mathtt{0}) = 1 + 1 + 0 = 2$$$.

In the third test case, $$$f$$$ equals to $$$0$$$ for all $$$1 \leq i \leq j \leq 5$$$. Thus, the answer is $$$0$$$.

Now solve the problem and return the code.