C2. Guessing the Greatest (hard version)time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThe only difference between the easy and the hard version is the limit to the number of queries.This is an interactive problem.There is an array a of n different numbers. In one query you can ask the position of the second maximum element in a subsegment a[l..r]. Find the position of the maximum element in the array in no more than 20 queries.A subsegment a[l..r] is all the elements a_l, a_{l + 1}, ..., a_r. After asking this subsegment you will be given the position of the second maximum from this subsegment in the whole array.InputThe first line contains a single integer n (2 \leq n \leq 10^5) — the number of elements in the array.InteractionYou can ask queries by printing "? l r" (1 \leq l < r \leq n). The answer is the index of the second maximum of all elements a_l, a_{l + 1}, ..., a_r. Array a is fixed beforehand and can't be changed in time of interaction.You can output the answer by printing "! p", where p is the index of the maximum element in the array.You can ask no more than 20 queries. Printing the answer doesn't count as a query.After printing a query do not forget to output end of line and flush the output. Otherwise, you will get Idleness limit exceeded. To do this, use:  fflush(stdout) or cout.flush() in C++;  System.out.flush() in Java;  flush(output) in Pascal;  stdout.flush() in Python;  see documentation for other languages HacksTo make a hack, use the following test format.In the first line output a single integer n (2 \leq n \leq 10^5). In the second line output a permutation of n integers 1 to n. The position of n in the permutation is the position of the maximumExampleInput
5

3

4

Output
? 1 5

? 4 5

! 1NoteIn the sample suppose a is [5, 1, 4, 2, 3]. So after asking the [1..5] subsegment 4 is second to max value, and it's position is 3. After asking the [4..5] subsegment 2 is second to max value and it's position in the whole array is 4.Note that there are other arrays a that would produce the same interaction, and the answer for them might be different. Example output is given in purpose of understanding the interaction.
5

3

4


? 1 5  ? 4 5  ! 1
['binary search', 'interactive', '*1900']