CF1442B.Identify the Operations

We start with a permutation $a_1, a_2, \ldots, a_n$ and with an empty array $b$ . We apply the following operation $k$ times.

On the $i$ -th iteration, we select an index $t_i$ ( $1 \le t_i \le n-i+1$ ), remove $a_{t_i}$ from the array, and append one of the numbers $a_{t_i-1}$ or $a_{t_i+1}$ (if $t_i-1$ or $t_i+1$ are within the array bounds) to the right end of the array $b$ . Then we move elements $a_{t_i+1}, \ldots, a_n$ to the left in order to fill in the empty space.

You are given the initial permutation $a_1, a_2, \ldots, a_n$ and the resulting array $b_1, b_2, \ldots, b_k$ . All elements of an array $b$ are distinct. Calculate the number of possible sequences of indices $t_1, t_2, \ldots, t_k$ modulo $998\,244\,353$ .

Each test contains multiple test cases. The first line contains an integer $t$ ( $1 \le t \le 100\,000$ ), denoting the number of test cases, followed by a description of the test cases.

The first line of each test case contains two integers $n, k$ ( $1 \le k < n \le 200\,000$ ): sizes of arrays $a$ and $b$ .

The second line of each test case contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le n$ ): elements of $a$ . All elements of $a$ are distinct.

The third line of each test case contains $k$ integers $b_1, b_2, \ldots, b_k$ ( $1 \le b_i \le n$ ): elements of $b$ . All elements of $b$ are distinct.

The sum of all $n$ among all test cases is guaranteed to not exceed $200\,000$ .

For each test case print one integer: the number of possible sequences modulo $998\,244\,353$ .

\require{cancel}

Let's denote as $a_1 a_2 \ldots \cancel{a_i} \underline{a_{i+1}} \ldots a_n \rightarrow a_1 a_2 \ldots a_{i-1} a_{i+1} \ldots a_{n-1}$ an operation over an element with index $i$ : removal of element $a_i$ from array $a$ and appending element $a_{i+1}$ to array $b$ .

In the first example test, the following two options can be used to produce the given array $b$ :

• $1 2 \underline{3} \cancel{4} 5 \rightarrow 1 \underline{2} \cancel{3} 5 \rightarrow 1 \cancel{2} \underline{5} \rightarrow 1 2$ ; $(t_1, t_2, t_3) = (4, 3, 2)$ ;
• $1 2 \underline{3} \cancel{4} 5 \rightarrow \cancel{1} \underline{2} 3 5 \rightarrow 2 \cancel{3} \underline{5} \rightarrow 1 5$ ; $(t_1, t_2, t_3) = (4, 1, 2)$ .

In the second example test, it is impossible to achieve the given array no matter the operations used. That's because, on the first application, we removed the element next to $4$ , namely number $3$ , which means that it couldn't be added to array $b$ on the second step.

In the third example test, there are four options to achieve the given array $b$ :

• $1 4 \cancel{7} \underline{3} 6 2 5 \rightarrow 1 4 3 \cancel{6} \underline{2} 5 \rightarrow \cancel{1} \underline{4} 3 2 5 \rightarrow 4 3 \cancel{2} \underline{5} \rightarrow 4 3 5$ ;
• $1 4 \cancel{7} \underline{3} 6 2 5 \rightarrow 1 4 3 \cancel{6} \underline{2} 5 \rightarrow 1 \underline{4} \cancel{3} 2 5 \rightarrow 1 4 \cancel{2} \underline{5} \rightarrow 1 4 5$ ;
• $1 4 7 \underline{3} \cancel{6} 2 5 \rightarrow 1 4 7 \cancel{3} \underline{2} 5 \rightarrow \cancel{1} \underline{4} 7 2 5 \rightarrow 4 7 \cancel{2} \underline{5} \rightarrow 4 7 5$ ;
• $1 4 7 \underline{3} \cancel{6} 2 5 \rightarrow 1 4 7 \cancel{3} \underline{2} 5 \rightarrow 1 \underline{4} \cancel{7} 2 5 \rightarrow 1 4 \cancel{2} \underline{5} \rightarrow 1 4 5$ ;