Description:
By the age of three Smart Beaver mastered all arithmetic operations and got this summer homework from the amazed teacher:
You are given a sequence of integers a1, a2, ..., an. Your task is to perform on it m consecutive operations of the following type:
1. For given numbers xi and vi assign value vi to element axi.
2. For given numbers li and ri you've got to calculate sum $$\sum_{x=0}^{r_i-l_i} (f_x \cdot a_{l_i+x})$$, where f0 = f1 = 1 and at i ≥ 2: fi = fi - 1 + fi - 2.
3. For a group of three numbers li ri di you should increase value ax by di for all x (li ≤ x ≤ ri).
Smart Beaver planned a tour around great Canadian lakes, so he asked you to help him solve the given problem.
Input Format:
The first line contains two integers n and m (1 ≤ n, m ≤ 2·105) — the number of integers in the sequence and the number of operations, correspondingly. The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ 105). Then follow m lines, each describes an operation. Each line starts with an integer ti (1 ≤ ti ≤ 3) — the operation type:
- if ti = 1, then next follow two integers xi vi (1 ≤ xi ≤ n, 0 ≤ vi ≤ 105);
- if ti = 2, then next follow two integers li ri (1 ≤ li ≤ ri ≤ n);
- if ti = 3, then next follow three integers li ri di (1 ≤ li ≤ ri ≤ n, 0 ≤ di ≤ 105).
The input limits for scoring 30 points are (subproblem E1):
- It is guaranteed that n does not exceed 100, m does not exceed 10000 and there will be no queries of the 3-rd type.
The input limits for scoring 70 points are (subproblems E1+E2):
- It is guaranteed that there will be queries of the 1-st and 2-nd type only.
The input limits for scoring 100 points are (subproblems E1+E2+E3):
- No extra limitations.
Output Format:
For each query print the calculated sum modulo 1000000000 (109).
Note:
None