CF107D Crime Management

Zeyad wants to commit $ n $ crimes in Egypt and not be punished at the end. There are several types of crimes. For example, bribery is a crime but is not considered such when repeated twice. Therefore, bribery is not considered a crime when repeated an even number of times. Speeding is a crime, but is not considered such when repeated a number of times which is a multiple of five. More specifically, $ c $ conditions on crime repetitions are known. Each condition describes the crime type $ t_{i} $ and its multiplicity $ m_{i} $ . If the number of times Zeyad committed the crime $ t_{i} $ is a multiple of $ m_{i} $ , Zeyad will not be punished for crime $ t_{i} $ . Some crimes may be listed more than once. In this case fulfilling at least one condition for this crime is enough to not be punished for it. Of course, if for certain crime the number of times Zeyad committed it is zero, he is innocent with respect to this crime. Now Zeyad is interested in a number of ways he can commit exactly $ n $ crimes without any punishment. The order of commiting the crimes matters. More formally, two ways, sequences $ w1 $ and $ w2 $ , of committing $ n $ crimes are equal if $ w1_{i}=w2_{i} $ , for all $ 1

The first line contains two integers $ n $ and $ c $ ( $ 0

Output the number of different ways Zeyad can commit exactly $ n $ crimes with no punishment modulo $ 12345 $ .

In the first test case, the 16 ways are: AAAAA, AAABB, AABAB, AABBA, ABAAB, ABABA, ABBAA, BAAAB, BAABA, BABAA, BBAAA, ABBBB, BABBB, BBABB, BBBAB, BBBBA.