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G - 012 Inversion

Score : $600$ points

Problem Statement

You are given a sequence $A=(A_1,\ldots,A_N)$ of length $N$. Each element is $0$, $1$, or $2$.
Process $Q$ queries in order. Each query is of one of the following kinds:

• 1 L R: print the inversion number of the sequence $(A_L,\ldots,A_R)$.
• 2 L R S T U: for each $i$ such that $L\leq i \leq R$, if $A_i$ is $0$, replace it with $S$; if $A_i$ is $1$, replace it with $T$; if $A_i$ is $2$, replace it with $U$.

What is the inversion number?

The inversion number of a sequence $B = (B_1, \ldots, B_M)$ is the number of pairs of integers $(i, j)$ $(1 \leq i B_j$.

Constraints

• $1 \leq N \leq 10^5$
• $0 \leq A_i \leq 2$
• $1\leq Q\leq 10^5$
• In each query, $1\leq L \leq R \leq N$.
• In each query of the second kind, $0\leq S,T,U \leq 2$.
• All values in input are integers.

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Input

Input is given from Standard Input in the following format:

$N$ $Q$

$A_1$ $A_2$ $\ldots$ $A_N$

$\rm Query_1$

$\rm Query_2$

$\vdots$

$\rm Query_Q$

$\rm Query_i$ denotes the $i$-th query, which is in one of the following formats:

``` $1$ $L$ $R$ ``` ``` $2$ $L$ $R$ $S$ $T$ $U$ ```

Output

Print the responses to the queries of the first kind in the given order, separated by newlines.

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5 3
2 0 2 1 0
1 2 5
2 2 4 2 1 0
1 2 5

3
4

Initially, $A=(2,0,2,1,0)$.

• In the $1$-st query, print the inversion number $3$ of $(A_2,A_3,A_4,A_5)=(0,2,1,0)$.
• The $2$-nd query makes $A=(2,2,0,1,0)$.
• In the $3$-rd query, print the inversion number $4$ of $(A_2,A_3,A_4,A_5)=(2,0,1,0)$.

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3 3
0 1 2
1 1 1
2 1 3 0 0 0
1 1 3

0
0