Contest Duration: - (local time) (120 minutes)
F - Pass /

Time Limit: 2 sec / Memory Limit: 1024 MB

問題文

N 人のすぬけ君が 1 列に並んでいます。 長さ N の文字列 S が与えられ、前から i 人目のすぬけ君は Si 文字目が 0 のとき赤いボールを 2 つ、1 のとき赤いボールと青いボールを 1 つずつ、2 のとき青いボールを 2 つ持っています。

• ボールを 1 つ以上持っているすぬけ君は全員同時に、自分が持っているボールの中から 1 つを選び、前のすぬけ君に渡す。ただし、先頭のすぬけ君は、高橋君に渡す。
• 高橋君は、先頭のすぬけ君から受け取ったボールを、列の末尾に並べる。

制約

• 1 \leq |S| \leq 2000
• S0,1,2 からなる

入力

S


入力例 1

02


出力例 1

3


入力例 2

1210


出力例 2

55


入力例 3

12001021211100201020


出力例 3

543589959


Score : 900 points

Problem Statement

There are N Snukes lining up in a row. You are given a string S of length N. The i-th Snuke from the front has two red balls if the i-th character in S is 0; one red ball and one blue ball if the i-th character in S is 1; two blue balls if the i-th character in S is 2.

Takahashi has a sequence that is initially empty. Find the number of the possible sequences he may have after repeating the following procedure 2N times, modulo 998244353:

• Each Snuke who has one or more balls simultaneously chooses one of his balls and hand it to the Snuke in front of him, or hand it to Takahashi if he is the first Snuke in the row.
• Takahashi receives the ball and put it to the end of his sequence.

Constraints

• 1 \leq |S| \leq 2000
• S consists of 0,1 and 2.

Note that the integer N is not directly given in input; it is given indirectly as the length of the string S.

Input

Input is given from Standard Input in the following format:

S


Output

Print the number of the possible sequences Takahashi may have after repeating the procedure 2N times, modulo 998244353.

Sample Input 1

02


Sample Output 1

3


There are three sequences that Takahashi may have: rrbb, rbrb and rbbr, where r and b stand for red and blue balls, respectively.

Sample Input 2

1210


Sample Output 2

55


Sample Input 3

12001021211100201020


Sample Output 3

543589959