Rings
Contest: Codeforces Round 741 (Div. 2)
Difficulty rating: 1500
Tags: constructive algorithms, math

Somewhere in a parallel Middle-earth, when Saruman caught Frodo, he only found $$$n$$$ rings. And the $$$i$$$-th ring was either gold or silver. For convenience Saruman wrote down a binary string $$$s$$$ of $$$n$$$ characters, where the $$$i$$$-th character was 0 if the $$$i$$$-th ring was gold, and 1 if it was silver.

Saruman has a magic function $$$f$$$, which takes a binary string and returns a number obtained by converting the string into a binary number and then converting the binary number into a decimal number. For example, $$$f(001010) = 10, f(111) = 7, f(11011101) = 221$$$.

Saruman, however, thinks that the order of the rings plays some important role. He wants to find $$$2$$$ pairs of integers $$$(l_1, r_1), (l_2, r_2)$$$, such that:

- $$$1 \le l_1 \le n$$$, $$$1 \le r_1 \le n$$$, $$$r_1-l_1+1\ge \lfloor \frac{n}{2} \rfloor$$$
- $$$1 \le l_2 \le n$$$, $$$1 \le r_2 \le n$$$, $$$r_2-l_2+1\ge \lfloor \frac{n}{2} \rfloor$$$
- Pairs $$$(l_1, r_1)$$$ and $$$(l_2, r_2)$$$ are distinct. That is, at least one of $$$l_1 \neq l_2$$$ and $$$r_1 \neq r_2$$$ must hold.
- Let $$$t$$$ be the substring $$$s[l_1:r_1]$$$ of $$$s$$$, and $$$w$$$ be the substring $$$s[l_2:r_2]$$$ of $$$s$$$. Then there exists non-negative integer $$$k$$$, such that $$$f(t) = f(w) \cdot k$$$.

Here substring $$$s[l:r]$$$ denotes $$$s_ls_{l+1}\ldots s_{r-1}s_r$$$, and $$$\lfloor x \rfloor$$$ denotes rounding the number down to the nearest integer.

Help Saruman solve this problem! It is guaranteed that under the constraints of the problem at least one solution exists.

Input format:
Each test contains multiple test cases.

The first line contains one positive integer $$$t$$$ ($$$1 \le t \le 10^3$$$), denoting the number of test cases. Description of the test cases follows.

The first line of each test case contains one positive integer $$$n$$$ ($$$2 \le n \le 2 \cdot 10^4$$$) — length of the string.

The second line of each test case contains a non-empty binary string of length $$$n$$$.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.

Output format:
For every test case print four integers $$$l_1$$$, $$$r_1$$$, $$$l_2$$$, $$$r_2$$$, which denote the beginning of the first substring, the end of the first substring, the beginning of the second substring, and the end of the second substring, respectively.

If there are multiple solutions, print any.

Constraints:
Time limit: 1.0 s
Memory limit: 256.0 MB

Note:
In the first testcase $$$f(t) = f(1111) = 15$$$, $$$f(w) = f(101) = 5$$$.

In the second testcase $$$f(t) = f(111000111) = 455$$$, $$$f(w) = f(000111) = 7$$$.

In the third testcase $$$f(t) = f(0000) = 0$$$, $$$f(w) = f(1000) = 8$$$.

In the fourth testcase $$$f(t) = f(11011) = 27$$$, $$$f(w) = f(011) = 3$$$.

In the fifth testcase $$$f(t) = f(001111) = 15$$$, $$$f(w) = f(011) = 3$$$.

Example input:
7
6
101111
9
111000111
8
10000000
5
11011
6
001111
3
101
30
100000000000000100000000000000

Example output:
3 6 1 3
1 9 4 9
5 8 1 4
1 5 3 5
1 6 2 4
1 2 2 3
1 15 16 30

Editorial:
Let us first consider the boundary case. Let a string (hereafter we will assume that the string length $$$n$$$) consists of only ones. Then we can output the numbers $$$1$$$ $$$n-1$$$ $$$2$$$ $$$n$$$ as the answer, since there will be the same substrings.
Now let's figure out what to do in the other case. Let's call the substring [$$$1$$$ ... $$$\lfloor\frac{n}{2}\rfloor$$$] the left half of the string, and the substring [$$$\lfloor\frac{n}{2}\rfloor+1$$$ ... $$$n$$$] the right half of the string. Then there are two cases:
• There is $$$0$$$ in the left half of the row, and its position is $$$k$$$. Then we can take the numbers $$$k$$$ $$$n$$$ $$$k+1$$$ $$$n$$$ as the answer, since they are the same numbers, just the second number has an extra leading zero.
• There is $$$0$$$ in the right half of the row, and its position is $$$k$$$. Then you can take the numbers $$$1$$$ $$$k$$$ $$$1$$$ $$$k-1$$$ as the answer, since the second number — is the first number multiplied by two (multiplying by two in binary is equivalent to adding one zero to the right).
Asymptotics: $$$O(n)$$$ per test case.