Integers and Ranges

Time limit: 3000 ms. Memory limit: 512 MB.

Isaac has a decimal integer $\overline{a_1 a_2 \dots a_n}$, possibly with leading zeroes. He knows that for $m$ ranges $[l_1, r_1], [l_2, r_2], \dots, [l_m, r_m]$, the product $a_{l_i} \times a_{l_i + 1} \times \dots \times a_{r_i} \bmod 9 = 0$ holds for each range $i$. Find the number of valid integers $\overline{a_1 a_2 \dots a_n}$, modulo $(10^9+7)$.

The input consists of several test cases and ends at end-of-file.

The first line of each test case contains two integers $n$ and $m$. The next $m$ lines each contain two integers $l_i$ and $r_i$.

For each test case, print one integer, the result.

• $1 \leq n, m \leq 10^3$
• $1 \leq l_i \leq r_i \leq n$
• There are at most $100$ test cases.