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E - Rem of Sum is Num

Score : $500$ points

Problem Statement

Given are a sequence of $N$ positive integers $A_1, A_2, \ldots, A_N$, and a positive integer $K$.

Find the number of non-empty contiguous subsequences in $A$ such that the remainder when dividing the sum of its elements by $K$ is equal to the number of its elements. We consider two subsequences different if they are taken from different positions, even if they are equal sequences.

Constraints

• All values in input are integers.
• $1 \leq N \leq 2\times 10^5$
• $1 \leq K \leq 10^9$
• $1 \leq A_i \leq 10^9$

Input

Input is given from Standard Input in the following format:

$N$ $K$

$A_1$ $A_2$ $\cdots$ $A_N$

Output

Print the number of subsequences that satisfy the condition.

5 4
1 4 2 3 5

4

Four sequences satisfy the condition: $(1)$, $(4,2)$, $(1,4,2)$, and $(5)$.

8 4
4 2 4 2 4 2 4 2

7

$(4,2)$ is counted four times, and $(2,4)$ is counted three times.

10 7
14 15 92 65 35 89 79 32 38 46

8