1 #include <cstdio>
 2 #include <iostream>
 3 #include <cmath>
 4 #include <cstring>
 5 using namespace std;
 6 struct edge {int l,r;}e[100010];
 7 int n,m,a[100010],cnt,feng;
 8 long long sum[100010],k[100010],tot[100010],f[100010],ans,mo=1e9+9;
 9 int main()
10 {
11     scanf("%d%d",&n,&m); feng=sqrt(m);
12     for (int i=1;i<=n;i++) scanf("%d",&a[i]),sum[i]=(sum[i-1]+a[i])%mo;
13     f[1]=tot[1]=1;
14     for (int i=2;i<=n;i++) f[i]=(f[i-1]+f[i-2])%mo,tot[i]=(tot[i-1]+f[i])%mo;
15     for (int j=1;j<=m;j++)
16     {
17         int d,l,r;
18         scanf("%d%d%d",&d,&l,&r);
19         if (d==1) e[++cnt].l=l,e[cnt].r=r; 
20         else 
21         {
22             ans=0;
23             for (int i=1;i<=cnt;i++)
24             {
25                 int a=e[i].l,b=e[i].r;
26                 if (r<a||l>b) continue;
27                 ans=(ans+tot[min(b,r)-(a-1)]-tot[max(a,l)-a]+mo)%mo;
28             }
29             printf("%lld\n",(ans+sum[r]-sum[l-1]+mo)%mo);
30         }
31         if (cnt==feng)
32         {
33             memset(k,0,sizeof(k));
34             for (int i=1;i<=cnt;i++)
35             {
36                 int l=e[i].l,r=e[i].r;
37                 k[l]=(k[l]+1)%mo;
38                 k[r+1]=(k[r+1]-f[r-l+2]+mo)%mo,k[r+2]=(k[r+2]-f[r-l+1]+mo)%mo;
39             }
40             cnt=0;
41             for (int i=1;i<=n;i++)
42             {
43                 k[i]=(k[i]+k[i-1]+k[i-2])%mo;
44                 a[i]=(a[i]+k[i])%mo;
45                 sum[i]=(sum[i-1]+a[i])%mo;
46             }
47         }
48     }
49 }