CF1868C.Travel Plan

During the summer vacation after Zhongkao examination, Tom and Daniel are planning to go traveling.

There are $n$ cities in their country, numbered from $1$ to $n$ . And the traffic system in the country is very special. For each city $i$ ( $1 \le i \le n$ ), there is

• a road between city $i$ and $2i$ , if $2i\le n$ ;
• a road between city $i$ and $2i+1$ , if $2i+1\le n$ .

Making a travel plan, Daniel chooses some integer value between $1$ and $m$ for each city, for the $i$ -th city we denote it by $a_i$ .

Let $s_{i,j}$ be the maximum value of cities in the simple $^\dagger$ path between cities $i$ and $j$ . The score of the travel plan is $\sum_{i=1}^n\sum_{j=i}^n s_{i,j}$ .

Tom wants to know the sum of scores of all possible travel plans. Daniel asks you to help him find it. You just need to tell him the answer modulo $998\,244\,353$ .

$^\dagger$ A simple path between cities $x$ and $y$ is a path between them that passes through each city at most once.

The first line of input contains a single integer $t$ ( $1\le t\le 200$ ) — the number of test cases. The description of test cases follows.

The only line of each test case contains two integers $n$ and $m$ ( $1\leq n\leq 10^{18}$ , $1\leq m\leq 10^5$ ) — the number of the cities and the maximum value of a city.

It is guaranteed that the sum of $m$ over all test cases does not exceed $10^5$ .

For each test case output one integer — the sum of scores of all possible travel plans, modulo $998\,244\,353$ .

In the first test case, there is only one possible travel plan:

Path $1\rightarrow 1$ : $s_{1,1}=a_1=1$ .

Path $1\rightarrow 2$ : $s_{1,2}=\max(1,1)=1$ .

Path $1\rightarrow 3$ : $s_{1,3}=\max(1,1)=1$ .

Path $2\rightarrow 2$ : $s_{2,2}=a_2=1$ .

Path $2\rightarrow 1\rightarrow 3$ : $s_{2,3}=\max(1,1,1)=1$ .

Path $3\rightarrow 3$ : $s_{3,3}=a_3=1$ .

The score is $1+1+1+1+1+1=6$ .

In the second test case, there are four possible travel plans:

Score of plan $1$ : $1+1+1=3$ .

Score of plan $2$ : $1+2+2=5$ .

Score of plan $3$ : $2+2+1=5$ .

Score of plan $4$ : $2+2+2=6$ .

Therefore, the sum of score is $3+5+5+6=19$ .