CF1808E3 Minibuses on Venus (hard version)

This is the hard version of the problem. The only difference between the three versions is the constraints on $ n $ and $ k $ . You can make hacks only if all versions of the problem are solved. Maxim is a minibus driver on Venus. To ride on Maxim's minibus, you need a ticket. Each ticket has a number consisting of $ n $ digits. However, as we know, the residents of Venus use a numeral system with base $ k $ , rather than the decimal system. Therefore, the ticket number can be considered as a sequence of $ n $ integers from $ 0 $ to $ k-1 $ , inclusive. The residents of Venus consider a ticket to be lucky if there is a digit on it that is equal to the sum of the remaining digits, modulo $ k $ . For example, if $ k=10 $ , then the ticket $ 7135 $ is lucky because $ 7 + 1 + 5 \equiv 3 \pmod{10} $ . On the other hand, the ticket $ 7136 $ is not lucky because no digit is equal to the sum of the others modulo $ 10 $ . Once, while on a trip, Maxim wondered: how many lucky tickets exist? At the same time, Maxim understands that this number can be very large, so he is interested only in the answer modulo some prime number $ m $ . 

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In the first example, there are only four lucky tickets: $ 000 $ , $ 011 $ , $ 101 $ , and $ 110 $ .