CF2170E Binary Strings and Blocks

Define a block in a binary string (a string consisting of characters 0 and/or 1) as its continuous substring of characters of the same type that cannot be extended either to the left or to the right. For example, in the string 110001111, there are three blocks: - 11 (from the $ 1 $ -st character to the $ 2 $ -nd character); - 000 (from the $ 3 $ -rd character to the $ 5 $ -th character); - 1111 (from the $ 6 $ -th character to the $ 9 $ -th character). The substring from the $ 7 $ -th character to the $ 9 $ -th character 111 is not a block because it can be extended to the left. The substring from the $ 1 $ -st character to the $ 5 $ -th character 11000 is not a block because it contains characters of different types. We call a string beautiful if we can remove exactly one block from it so that the resulting string consists of an odd number of blocks. For example: - the string 110001111 is beautiful because we can remove the block from the $ 3 $ -rd to the $ 5 $ -th character, resulting in the string 111111, which consists of one block; - the string 1010 is beautiful because we can remove the block from the $ 1 $ -st to the $ 1 $ -st character, resulting in the string 010, which consists of three blocks; - the string 0000 is not beautiful because the only way to remove a block from it will result in an empty string, which consists of $ 0 $ blocks. You are given an integer $ n $ and $ m $ constraints, the $ i $ -th of which is described by a pair of integers $ l_i, r_i $ . We denote the substring of the string $ s $ from character $ l $ to character $ r $ inclusive as $ s[l:r] $ , that is, $ s[l:r] = s_l s_{l+1} \dots s_r $ . Your task is to count the number of binary strings $ s $ of length $ n $ meeting the following condition: - for each $ i $ from $ 1 $ to $ m $ , the string $ s[l_i:r_i] $ is beautiful.

The first line contains one integer $ t $ ( $ 1 \le t \le 10^4 $ ) — the number of test case. The first line of each test case contains two integers $ n $ and $ m $ ( $ 2 \le n \le 3 \cdot 10^5 $ ; $ 1 \le m \le 3 \cdot 10^5 $ ) — the required length of the string and the number of constraints, respectively. Next, there are $ m $ lines, the $ i $ -th of which contains two integers $ l_i, r_i $ ( $ 1 \le l_i < r_i \le n $ ) — the description of the $ i $ -th constraint. Additional constraints on the input: - The sum of $ n $ across all test cases does not exceed $ 3 \cdot 10^5 $ ; - The sum of $ m $ across all test cases does not exceed $ 3 \cdot 10^5 $ .

For each test case, print one integer — the number of strings that satisfy the condition. Since it may be huge, output it modulo $ 998244353 $ .

In the first example from the statement, the following strings are suitable: 1010, 0101. For each of these strings, both $ s[1:2] $ , $ s[2:3] $ , and $ s[3:4] $ are beautiful.