Balanced Seesaw Array

Bob likes playing on a seesaw. He thinks it is really fun when the seesaw is balanced, meaning it does not tilt left or right. After playing, Bob thinks about a problem involving balanced seesaw arrays.

Let $A = [a_1, a_2, \dots, a_m]$ be an array of length $m$. $A$ is a balanced seesaw array if there exists an integer $k$ with $1 \le k \le m$ such that $\sum_{i=1}^{m}(i-k)a_i = 0$.

Bob received an array $A = [a_1, a_2, \dots, a_n]$ as a birthday gift. He wants to know whether some non-empty subarray is a balanced seesaw array. Specifically, for given $(\ell, r)$ with $1 \le \ell \le r \le n$, he asks whether $[a_\ell, \dots, a_r]$ is a balanced seesaw array. The elements change over time through two kinds of updates:

1. Add $x$ to each of $a_\ell, \dots, a_r$.
2. Set each of $a_\ell, \dots, a_r$ to $x$.

Bob first gives you the array. Then there are $q$ operations, each of one of the following three types.

• 1 l r x: add $x$ to each of $a_\ell, \dots, a_r$.
• 2 l r x: set each of $a_\ell, \dots, a_r$ to $x$.
• 3 l r: check whether $[a_\ell, \dots, a_r]$ is a balanced seesaw array.

The first line contains two integers $n$ and $q$, the length of the array and the number of operations. The second line contains $n$ integers $a_i$. Each of the following $q$ lines describes one operation as defined above.

For each operation of type 3, print Yes if the subarray is a balanced seesaw array, otherwise print No.

• $1 \le n \le 100000$
• $1 \le q \le 1200000$
• $-1000 \le a_i \le 1000$
• $-10000 \le x \le 10000$
• For $1 \le i \le n$, you may assume that $|a_i| \le 1.5 \times 10^9$ after any operation.
• $1 \le \ell \le r \le n$