CF1004F.Sonya and Bitwise OR

Sonya has an array $a_1, a_2, \ldots, a_n$ consisting of $n$ integers and also one non-negative integer $x$ . She has to perform $m$ queries of two types:

• $1$ $i$ $y$ : replace $i$ -th element by value $y$ , i.e. to perform an operation $a_{i}$ := $y$ ;
• $2$ $l$ $r$ : find the number of pairs ( $L$ , $R$ ) that $l\leq L\leq R\leq r$ and bitwise OR of all integers in the range $[L, R]$ is at least $x$ (note that $x$ is a constant for all queries).

Can you help Sonya perform all her queries?

Bitwise OR is a binary operation on a pair of non-negative integers. To calculate the bitwise OR of two numbers, you need to write both numbers in binary notation. The result is a number, in binary, which contains a one in each digit if there is a one in the binary notation of at least one of the two numbers. For example, $10$ OR $19$ = $1010_2$ OR $10011_2$ = $11011_2$ = $27$ .

The first line contains three integers $n$ , $m$ , and $x$ ( $1\leq n, m\leq 10^5$ , $0\leq x<2^{20}$ ) — the number of numbers, the number of queries, and the constant for all queries.

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $0\leq a_i<2^{20}$ ) — numbers of the array.

The following $m$ lines each describe an query. A line has one of the following formats:

• $1$ $i$ $y$ ( $1\leq i\leq n$ , $0\leq y<2^{20}$ ), meaning that you have to replace $a_{i}$ by $y$ ;
• $2$ $l$ $r$ ( $1\leq l\leq r\leq n$ ), meaning that you have to find the number of subarrays on the segment from $l$ to $r$ that the bitwise OR of all numbers there is at least $x$ .

For each query of type 2, print the number of subarrays such that the bitwise OR of all the numbers in the range is at least $x$ .

In the first example, there are an array [ $0$ , $3$ , $6$ , $1$ ] and queries:

1. on the segment [ $1\ldots4$ ], you can choose pairs ( $1$ , $3$ ), ( $1$ , $4$ ), ( $2$ , $3$ ), ( $2$ , $4$ ), and ( $3$ , $4$ );
2. on the segment [ $3\ldots4$ ], you can choose pair ( $3$ , $4$ );
3. the first number is being replacing by $7$ , after this operation, the array will consist of [ $7$ , $3$ , $6$ , $1$ ];
4. on the segment [ $1\ldots4$ ], you can choose pairs ( $1$ , $1$ ), ( $1$ , $2$ ), ( $1$ , $3$ ), ( $1$ , $4$ ), ( $2$ , $3$ ), ( $2$ , $4$ ), and ( $3$ , $4$ );
5. on the segment [ $1\ldots3$ ], you can choose pairs ( $1$ , $1$ ), ( $1$ , $2$ ), ( $1$ , $3$ ), and ( $2$ , $3$ );
6. on the segment [ $1\ldots1$ ], you can choose pair ( $1$ , $1$ );
7. the third number is being replacing by $0$ , after this operation, the array will consist of [ $7$ , $3$ , $0$ , $1$ ];
8. on the segment [ $1\ldots4$ ], you can choose pairs ( $1$ , $1$ ), ( $1$ , $2$ ), ( $1$ , $3$ ), and ( $1$ , $4$ ).

In the second example, there are an array [ $6$ , $0$ , $3$ , $15$ , $2$ ] are queries:

1. on the segment [ $1\ldots5$ ], you can choose pairs ( $1$ , $3$ ), ( $1$ , $4$ ), ( $1$ , $5$ ), ( $2$ , $4$ ), ( $2$ , $5$ ), ( $3$ , $4$ ), ( $3$ , $5$ ), ( $4$ , $4$ ), and ( $4$ , $5$ );
2. the fourth number is being replacing by $4$ , after this operation, the array will consist of [ $6$ , $0$ , $3$ , $4$ , $2$ ];
3. on the segment [ $1\ldots5$ ], you can choose pairs ( $1$ , $3$ ), ( $1$ , $4$ ), ( $1$ , $5$ ), ( $2$ , $4$ ), ( $2$ , $5$ ), ( $3$ , $4$ ), and ( $3$ , $5$ );
4. on the segment [ $3\ldots5$ ], you can choose pairs ( $3$ , $4$ ) and ( $3$ , $5$ );
5. on the segment [ $1\ldots4$ ], you can choose pairs ( $1$ , $3$ ), ( $1$ , $4$ ), ( $2$ , $4$ ), and ( $3$ , $4$ ).