Bash and a Tough Math Puzzle

题意翻译

给定一个数列$a_1,a_2,...,a_n$,支持两种操作 1 l r x,猜测数列中[l,r]位置上的数的最大公约数xx,判断这个猜测是否是接近正确的。如果我们可以在数列[l,r]位置中改动至多一个数使得它们的最大公约数是x,那么这个猜测就被认为是接近正确的(注意我们不需要在数列中进行实际的改动)。如果这个猜测是接近正确的,输出"YES",否则输出"NO"(都不含引号)。 2 i y,将a_i的数值改为y。

题目描述

Bash likes playing with arrays. He has an array $ a_{1},a_{2},...\ a_{n} $ of $ n $ integers. He likes to guess the greatest common divisor (gcd) of different segments of the array. Of course, sometimes the guess is not correct. However, Bash will be satisfied if his guess is almost correct. Suppose he guesses that the gcd of the elements in the range $ [l,r] $ of $ a $ is $ x $ . He considers the guess to be almost correct if he can change at most one element in the segment such that the gcd of the segment is $ x $ after making the change. Note that when he guesses, he doesn't actually change the array — he just wonders if the gcd of the segment can be made $ x $ . Apart from this, he also sometimes makes changes to the array itself. Since he can't figure it out himself, Bash wants you to tell him which of his guesses are almost correct. Formally, you have to process $ q $ queries of one of the following forms: - $ 1lrx $ — Bash guesses that the gcd of the range $ [l,r] $ is $ x $ . Report if this guess is almost correct. - $ 2iy $ — Bash sets $ a_{i} $ to $ y $ . Note: The array is $ 1 $ -indexed.

输入输出格式

输入格式


The first line contains an integer $ n $ ( $ 1<=n<=5·10^{5} $ ) — the size of the array. The second line contains $ n $ integers $ a_{1},a_{2},...,a_{n} $ ( $ 1<=a_{i}<=10^{9} $ ) — the elements of the array. The third line contains an integer $ q $ ( $ 1<=q<=4·10^{5} $ ) — the number of queries. The next $ q $ lines describe the queries and may have one of the following forms: - $ 1lrx $ ( $ 1<=l<=r<=n,1<=x<=10^{9} $ ). - $ 2iy $ $ (1<=i<=n,1<=y<=10^{9}) $ . Guaranteed, that there is at least one query of first type.

输出格式


For each query of first type, output "YES" (without quotes) if Bash's guess is almost correct and "NO" (without quotes) otherwise.

输入输出样例

输入样例 #1

3
2 6 3
4
1 1 2 2
1 1 3 3
2 1 9
1 1 3 2

输出样例 #1

YES
YES
NO

输入样例 #2

5
1 2 3 4 5
6
1 1 4 2
2 3 6
1 1 4 2
1 1 5 2
2 5 10
1 1 5 2

输出样例 #2

NO
YES
NO
YES

说明

In the first sample, the array initially is $ {2,6,3} $ . For query $ 1 $ , the first two numbers already have their gcd as $ 2 $ . For query $ 2 $ , we can achieve a gcd of $ 3 $ by changing the first element of the array to $ 3 $ . Note that the changes made during queries of type $ 1 $ are temporary and do not get reflected in the array. After query $ 3 $ , the array is now $ {9,6,3} $ . For query $ 4 $ , no matter which element you change, you cannot get the gcd of the range to be $ 2 $ .