E3. Summer Homework
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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 a 1, a 2, ..., a n. Your task is to perform on it m consecutive operations of the following type:

1. For given numbers x i and v i assign value v i to element a x i.
2. For given numbers l i and r i you've got to calculate sum , where f 0 = f 1 = 1 and at i ≥ 2: f i = f i - 1 + f i - 2.
3. For a group of three numbers l i r i d i you should increase value a x by d i for all x (l i ≤ x ≤ r i).

Smart Beaver planned a tour around great Canadian lakes, so he asked you to help him solve the given problem.

Input

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 a 1, a 2, ..., a n (0 ≤ a i ≤ 105). Then follow m lines, each describes an operation. Each line starts with an integer t i (1 ≤ t i ≤ 3) — the operation type:

• if t i = 1, then next follow two integers x i v i (1 ≤ x i ≤ n, 0 ≤ v i ≤ 105);
• if t i = 2, then next follow two integers l i r i (1 ≤ l i ≤ r i ≤ n);
• if t i = 3, then next follow three integers l i r i d i (1 ≤ l i ≤ r i ≤ n, 0 ≤ d i ≤ 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

For each query print the calculated sum modulo 1000000000 (109).

Examples
Input
5 51 3 1 2 42 1 42 1 52 2 41 3 102 1 5
Output
1232850
Input
5 41 3 1 2 43 1 4 12 2 41 2 102 1 5
Output
1245