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4 초 512 MB555100.000%

문제

Three friends are studying random walks. To delve deeper into the topic, they have decided to play a game.

Initially, the friends stand at integer points $x_1, x_2, x_3$ on the number line.

The game lasts for $k$ seconds.

Each second, an integer $j$ is chosen uniformly at random from the set $\{ 1, 2, 3 \}$. Then, friend $j$ increases their coordinate by $1$ with probability $p$ percent, or decreases their coordinate by $1$ with probability $(100 - p)$ percent.

Note that multiple friends can stand at the same point, both initially and during the game.

The stretch is defined as the length of the shortest segment on the number line containing all three friends.

Find the expected stretch after $k$ seconds, modulo $998\,244\,353$ (see the Output section for details).

입력

The first line contains three integers $x_1$, $x_2$, and $x_3$ ($-10^5 \le x_i \le 10^5$).

The second line contains a single integer $k$ ($1 \le k \le 2 \cdot 10^5$).

The third line contains a single integer $p$ ($0 \le p \le 100$).

출력

Print the expected stretch after $k$ seconds, modulo $998\,244\,353$.

Formally, let $M = 998\,244\,353$. It can be shown that the required expected stretch can be expressed as an irreducible fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \not \equiv 0 \pmod{M}$. Print the integer equal to $p \cdot q^{-1} \bmod M$. In other words, print such an integer $x$ that $0 \le x < M$ and $x \cdot q \equiv p \pmod{M}$.

예제 입력 1

0 0 0
1
58

예제 출력 1

1

예제 입력 2

1 2 2
1
100

예제 출력 2

332748119

예제 입력 3

5 2 3
4
50

예제 출력 3

160212060

힌트

In the first example test, regardless of which friend and direction are chosen, the stretch will be equal to $1$.

In the second example test, the actual expected stretch is $\frac{4}{3}$.

In the third example test, the actual expected stretch is $\frac{271}{81}$.