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No.3682 きあいのハチマキ

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作問者 : Andrew8128 / テスター : drken1215 ゆにたりー卿 Mikan04y lp_ql yimiya（いみや） よには 👑 loop0919 elphe edon8618 👑 seekworser leaf_1415 紫前燈太 triangle_coder 👑 みうね
お気に入りにしたユーザー ProblemId : 13427 / 自分の提出
問題文最終更新日: 2026-09-05 09:49:42
Focus Band
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第3回 岩井星人アンソロジープログラミングコンテスト (順位表) の他の問題:

Problem Statement

Cyan-rated Iwai Seijin and Green-rated Iwai Seijin are having a battle.
Cyan-rated Iwai Seijin has initial HP $H_C$, attack power $A_C$, and speed $S_C$.
Green-rated Iwai Seijin has initial HP $H_G$, attack power $A_G$, and speed $S_G$.

The battle proceeds as follows.

    • If the speeds of Cyan-rated Iwai Seijin and Green-rated Iwai Seijin differ, the Iwai Seijin with the greater speed attacks (the other Iwai Seijin receives the attack).
    • If the speeds of Cyan-rated Iwai Seijin and Green-rated Iwai Seijin are the same, Cyan-rated Iwai Seijin attacks with probability $\frac{1}{2}$, and Green-rated Iwai Seijin attacks with probability $\frac{1}{2}$.
  2. The Iwai Seijin who did not attack in step 1. attacks.
  3. Return to step 1.

When an attack is made, the HP of the Iwai Seijin receiving the attack decreases by the attack power of the Iwai Seijin making the attack.

Both Iwai Seijin hold a Focus Band, so when an attack would bring their HP to $0$ or below, with probability $\frac{1}{10}$ they endure it and remain with $1$ HP.

Note that when they receive an attack while at $1$ HP, they also endure it and remain with $1$ HP with probability $\frac{1}{10}$.

More formally, when an Iwai Seijin with current HP $H$ receives an attack from an Iwai Seijin with attack power $A$:
  • If $H \gt A$, the new HP of the Iwai Seijin receiving the attack becomes $H - A$.
  • If $H \leq A$, the new HP of the Iwai Seijin receiving the attack becomes $1$ with probability $\frac{1}{10}$, and $H - A$ with probability $\frac{9}{10}$.

The battle ends immediately as soon as either one's HP becomes $0$ or below, and the Iwai Seijin whose HP became $0$ or below loses. (The Iwai Seijin who did not lose wins.)

Find the probability that Cyan-rated Iwai Seijin has won when the battle ends, modulo $\mathrm{mod}\ 998244353$.

What is a probability $\mathrm{mod}\ 998244353$? It can be proved that the desired probability is always a rational number. It can also be proved that under the constraints of this problem, when the value is expressed as an irreducible fraction $\frac{P}{Q}$, we have $Q \not\equiv 0\ (\mathrm{mod}\ 998244353)$. Therefore, there is a unique integer $R$ satisfying $ R \times Q \equiv P\ (\mathrm{mod}\ 998244353), 0 \leq R \lt 998244353$. Report this $R$.

You are given $T$ test cases; find the answer for each of them.

Constraints

    All input values are integers.
    $1 \leq T \leq 2 \times 10^{4}$
    $1 \leq H_C, A_C, S_C, H_G, A_G, S_G \leq 10^{18}$

Input

The input is given from standard input in the following format, where $\mathrm{case}_i$ denotes the $i$-th test case.

$T$
$\mathrm{case}_1$
$\mathrm{case}_2$
$\vdots$
$\mathrm{case}_T$

Each test case is given in the following format.

$H_C\ A_C\ S_C\ H_G\ A_G\ S_G$

Output

Output $T$ lines.
On the $i$-th line, output the answer to the $i$-th test case.

Sample

Sample 1
Input
5
2 1 1 1 1 2
1000 100 10 1000 100 10
31 4 1 5 9 2
1000000000000000000 1000000000000000000 1000000000000000000 1000000000000000000 2408866010 1000000000000000000
819558960 819558960 819558960 1 1 1
Output
725995894
499122177
971019508
0
20260516

In the first test case, the probability that Cyan-rated Iwai Seijin wins is $\frac{10}{11}$. In the second test case, the probability that Cyan-rated Iwai Seijin wins is $\frac{1}{2}$.

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