You, a wizard whose creation was destroyed by a dragon, are determined to hunt it down with a magical AOE tracker. But it seems to be toyed with...

This is an interactive problem.

There is a hidden binary array $$$a$$$ of length $$$n$$$ ($$$\mathbf{n}$$$ is a power of 2) and a hidden integer $$$k\ (2 \le k \le n - 1)$$$. The array $$$a$$$ contains exactly one 1 (and all other elements are 0). For two integers $$$l$$$ and $$$r$$$ ($$$1 \le l \le r \le n$$$), define the range sum $$$s(l, r) = a_l + a_{l+1} + \cdots + a_r$$$.

You have a magical device that takes ranges and returns range sums, but it returns the opposite result when the range has length at least $$$k$$$. Formally, in one query, you can give it a pair of integers $$$[l, r]$$$ where $$$1 \le l \le r \le n$$$, and it will return either $$$0$$$ or $$$1$$$ according to the following rules:

- If $$$r - l + 1 < k$$$, it will return $$$s(l, r)$$$.
- If $$$r - l + 1 \ge k$$$, it will return $$$1 - s(l, r)$$$.

Find $$$k$$$ using at most $$$33$$$ queries.

The device is not adaptive. It means that the hidden $$$a$$$ and $$$k$$$ are fixed before the interaction and will not change during the interaction.

Example Input 0:
2
8

0

0

1

0

4

1

0

Example Output 0:
? 3 5

? 1 8

? 4 8

? 3 8

! 6

? 3 3

? 3 4

! 2

Notes:
In the first test case, $$$k = 6$$$ and the 1 in the hidden array is at index 6, so $$$a = [0, 0, 0, 0, 0, 1, 0, 0]$$$.

- For the query 3 5, since $$$5-3+1 = 3 < k$$$, the device answers correctly. Since 6 is not contained in the range $$$[3, 5]$$$, the device answers $$$0$$$.
- For the query 1 8, since $$$8 - 1 + 1 = 8 \ge k$$$, the device answers $$$0$$$ incorrectly.
- For the query 4 8, since $$$8 - 4 + 1 = 5 < k$$$, the device answers $$$1$$$ correctly.
- For the query 3 8, since $$$8 - 3 + 1 = 6 \ge k$$$, the device answers $$$0$$$ incorrectly.

In the second test case, $$$k = 2$$$ and the 1 in the hidden array is at index 3, so $$$a = [0, 0, 1, 0]$$$.

Note that the example solution may not have enough information to determine $$$k$$$ above; this is only an example.

Example input:
2
8

0

0

1

0

4

1

0


Example output:
? 3 5

? 1 8

? 4 8

? 3 8

! 6

? 3 3

? 3 4

! 2