A16195. Make it Alternating

You are given a binary string $s$ . A binary string is a string consisting of characters 0 and/or 1.

You can perform the following operation on $s$ any number of times (even zero):

- choose an integer $i$ such that $1 \le i \le |s|$ , then erase the character $s_i$ .

You have to make $s$ alternating, i. e. after you perform the operations, every two adjacent characters in $s$ should be different.

Your goal is to calculate two values:

- the minimum number of operations required to make $s$ alternating;
- the number of different shortest sequences of operations that make $s$ alternating. Two sequences of operations are different if in at least one operation, the chosen integer $i$ is different in these two sequences.

The first line contains one integer $t$ ( $1 \le t \le 10^4$ ) — the number of test cases.

Each test case consists of one line containing the string $s$ ( $1 \le |s| \le 2 \cdot 10^5$ ). The string $s$ consists of characters 0 and/or 1 only.

Additional constraint on the input:

- the total length of strings $s$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, print two integers: the minimum number of operations you have to perform, and the number of different shortest sequences of operations. Since the second number might be large, print its remainder modulo $998244353$ .

In the first test case of the example, the shortest sequences of operations are:

- $[2]$ (delete the $2$ -nd character);
- $[3]$ (delete the $3$ -rd character).

In the second test case of the example, the shortest sequences of operations are:

- $[2, 1]$ (delete the $2$ -nd character, then delete the $1$ -st character);
- $[2, 2]$ ;
- $[1, 1]$ ;
- $[1, 2]$ ;
- $[3, 1]$ ;
- $[3, 2]$ .

In the third test case of the example, the only shortest sequence of operations is $[]$ (empty sequence).