This is the hard version of the problem. The difference between the versions is that in this version, the constraints on $$$k$$$ and $$$q$$$ are larger. You can hack only if you solved all versions of this problem.
Define $$$\\mathrm{popcount}_k(m)$$$ as the sum of all digits of $$$m$$$ in base $$$k$$$.
Define a base-$$$k$$$ integer $$$s=\\overline{s_1s_2\\cdots}$$$ of infinite length, where the $$$i$$$-th digit of $$$s$$$ is $$$s_i=(\\mathrm{popcount}_k(i) \\bmod k)$$$.
There are $$$q$$$ queries. Each query consists of three integers $$$l$$$, $$$r$$$, and $$$n$$$ (all of them are given in decimal form), as well as a base-$$$k$$$ integer $$$t$$$ with $$$n$$$ digits. Note that $$$t$$$ may have leading zeros. Consider $$$t$$$ as a string, and your task is to find the number of occurrences of $$$t$$$ in the string $$$s_ls_{l+1}\\ldots s_r$$$.
For digits greater than or equal to decimal $$$\\mathtt{10}$$$, uppercase and lowercase letters are used. Specifically, the uppercase letters $$$\\{\\mathtt{A, B,\\ldots,Z}\\}$$$ represent the decimal values $$$\\{\\mathtt{10, 11,\\ldots,35}\\}$$$, and the lowercase letters $$$\\{\\mathtt{a, b,\\ldots,z}\\}$$$ represent the decimal values $$$\\{\\mathtt{36, 37,\\ldots,61}\\}$$$.
The first line of the input contains two integers $$$k$$$ and $$$q$$$ ($$$2\\le k\\le 62$$$, $$$1\\le q\\le 10^4$$$) — the base and the number of queries.
Each query contains two lines. The first line contains the three integers $$$l$$$, $$$r$$$, and $$$n$$$ ($$$1\\le l\\le r\\le 10^{17}$$$, $$$1\\le n\\le 2\\cdot 10^6$$$).
The second line contains the base-$$$k$$$ integer $$$t$$$ with $$$n$$$ digits ($$$t_i\\in\\{\\mathtt{0,1,\\ldots,9,A,B,\\ldots,Z,a,b,\\ldots,z}\\}$$$).
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$2 \\cdot 10^6$$$.
For each query, output a single integer — the answer to the query.
3 45 17 32015 17 2011239 1231231209 501201123002 231203 41202
2\n3\n21046666\n8325\n
10 31 20 912345678915 20332 36781234 56789 201
2\n1625\n5000\n
62 11 100000 620123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz
25\n
Denote the substring $$$s_ls_{l+1}\\ldots s_r$$$ as $$$s[l;r]$$$.
In the first example, $$$k = 3$$$, $$$s = \\mathtt{12120201120201012201120012\\ldots}$$$, $$$s[5;17] = \\mathtt{0201120201012}$$$, and $$$t = \\mathtt{201}$$$ appears a total of $$$2$$$ times in $$$s[5; 17]$$$. Their indices in the string $$$s$$$ are $$$s[6; 8]$$$ and $$$s[12; 14]$$$. And $$$t = \\mathtt{01}$$$ appears $$$3$$$ times. Their indices in the string $$$s$$$ are $$$s[7;8]$$$, $$$s[13;14]$$$, and $$$s[15;16]$$$.
For the second example, $$$k=10$$$, $$$s[1;20] = \\mathtt{12345678912345678902}$$$, and $$$t = \\mathtt{123456789}$$$ appears a total of $$$2$$$ times in $$$s[1;20]$$$.