“Multidimensional spaces are completely out of style these days, unlike genetics problems” — thought physicist Woll and changed his subject of study to bioinformatics. Analysing results of sequencing he faced the following problem concerning DNA sequences. We will further think of a DNA sequence as an arbitrary string of uppercase letters “A”, “C”, “G” and “T” (of course, this is a simplified interpretation).
Let $w$ be a long DNA sequence and $s_1, s_2, …, s_m$ — collection of short DNA sequences. Let us say that the collection filters $w$ iff $w$ can be covered with the sequences from the collection. Certainly, substrings corresponding to the different positions of the string may intersect or even cover each other. More formally: denote by $w$ the length of $w$, let symbols of $w$ be numbered from $1$ to $w$. Then for each position $i$ in $w$ there exist pair of indices $l$, $r$($1 ≤ l ≤ i ≤ r ≤ w$) such that the substring $w$$[l … r]$ equals one of the elements $s_1, s_2, …, s_m$ of the collection.
Woll wants to calculate the number of DNA sequences of a given length filtered by a given collection, but he doesn’t know how to deal with it. Help him! Your task is to find the number of different DNA sequences of length $n$ filtered by the collection {$s_i$}.
Answer may appear very large, so output it modulo $1000000009$.
Input
First line contains two integer numbers $n$ and $m$ ($1 ≤ n ≤ 1000$, $1 ≤ m ≤ 10$) — the length of the string and the number of sequences in the collection correspondently.
Next $m$ lines contain the collection sequences $s$$i$, one per line. Each $s_i$ is a nonempty string of length not greater than 10. All the strings consist of uppercase letters “A”, “C”, “G”, “T”. The collection may contain identical strings.
Output
Output should contain a single integer — the number of strings filtered by the collection modulo $1000000009$ ($10^9 + 9$).
Examples
input
1  2 1 
output
1  1 
input
1  6 2 
output
1  2 
Note
In the first sample, a string has to be filtered by “A”. Clearly, there is only one such string: “AA”.
In the second sample, there exist exactly two different strings satisfying the condition (see the pictures below).
题目大意
给你m个模式串，要求构造一个长度为n的串，要求串的每一位至少属于一个模式串。
题解
显然dp，最开始想了一个dp[i][j][0/1]表示当前位为j，还有i个位没填，该位是否位某个串的结尾，的方案数，写了个记忆化搜索，然后算重了。
dp[i][j][k]表示已经填了i位，结尾的一位在trie中标号为j，最后的k位目前没有属于任何一个模式串的方案数。
val[i]表示以trie数上的点i为结尾，能表示的串的最大的长度。
数组开小了2次，调了好久QAQ。
1 
