C. Make it Alternating
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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.
Input

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.

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

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 $998244353$.

Example
Input
3100101110101
Output
1 2
2 6
0 1

Note

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

• $$ (delete the $2$-nd character);
• $$ (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).