Codeforces Round #FF (Div. 2)E（线段树成段更新）

C. DZY Loves Fibonacci Numbers

time limit per test

4 seconds

memory limit per test

256 megabytes

input

standard input

output

standard output

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

F1 = 1; F2 = 1; Fn = Fn - 1 + Fn - 2 (n > 2).

DZY loves Fibonacci numbers very much. Today DZY gives you an array consisting of n integers: a1, a2, ..., an. Moreover, there are mqueries, each query has one of the two types:

1. Format of the query "1 l r". In reply to the query, you need to add Fi - l + 1 to each element ai, where l ≤ i ≤ r.
2. Format of the query "2 l r". In reply to the query you should output the value of  modulo 1000000009 (109 + 9).

Help DZY reply to all the queries.

Input

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 300000). The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — initial array a.

Then, m lines follow. A single line describes a single query in the format given in the statement. It is guaranteed that for each query inequality 1 ≤ l ≤ r ≤ n holds.

Output

For each query of the second type, print the value of the sum on a single line.

Sample test(s)

input

4 4
1 2 3 4
1 1 4
2 1 4
1 2 4
2 1 3

output

17
12

题意：支持两种操作，1 l r 将 ai 加上第i-l+1项Fibonacci数，l<=i<=r ，2 l r 求区间和

思路：根据Fibonacci数列的性质，只要知道前两项，那么可以O（1）得到后面任意项的Fibonacci数以及任意前多少项的和，可以用线段树维护每个线段的和以及每段的前两

            个Fibonacci数，延迟更新的是每个线段的前两个Fibonacci数（直接累加）