write a go solution for Description:
The 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 40 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.

Input Format:
The first line contains a single integer n (2<=n<=10^5) — the number of elements in the array.

Output Format:
None

Note:
In 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.. Output only the code with no comments, explanation, or additional text.