{"problem":{"name":"Broken Wheel","description":{"content":"You are given a positive integer $N$ and a length-$N$ string $s_0s_1\\ldots s_{N-1}$ consisting only of `0` and `1`. Consider a simple undirected graph $G$ with $(N+1)$ vertices numbered $0, 1, 2, \\ldo","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"arc193_b"},"statements":[{"statement_type":"Markdown","content":"You are given a positive integer $N$ and a length-$N$ string $s_0s_1\\ldots s_{N-1}$ consisting only of `0` and `1`.\nConsider a simple undirected graph $G$ with $(N+1)$ vertices numbered $0, 1, 2, \\ldots, N$, and the following edges:\n\n*   For each $i = 0, 1, \\ldots, N-1$, there is an undirected edge between vertices $i$ and $(i+1)\\bmod N$.\n*   For each $i = 0, 1, \\ldots, N-1$, there is an undirected edge between vertices $i$ and $N$ if and only if $s_i = $ `1`.\n*   There are no other edges.\n\nFurthermore, create a directed graph $G'$ by assigning a direction to each edge of $G$. That is, for each undirected edge $\\lbrace u, v \\rbrace$ in $G$, replace it with either a directed edge $(u, v)$ from $u$ to $v$ or a directed edge $(v, u)$ from $v$ to $u$.\nFor each $i = 0, 1, \\ldots, N$, let $d_i$ be the in-degree of vertex $i$ in $G'$. Print the number, modulo $998244353$, of distinct sequences $(d_0, d_1, \\ldots, d_N)$ that can be obtained.\n\n## Constraints\n\n*   $3 \\leq N \\leq 10^6$\n*   $N$ is an integer.\n*   Each $s_i$ is `0` or `1`.\n\n## Input\n\nThe input is given from Standard Input in the following format:\n\n$N$\n$s_0s_1\\ldots s_{N-1}$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"arc193_b","tags":[],"sample_group":[["3\n010","14\n\n$G$ has four undirected edges: $\\lbrace 0, 1 \\rbrace, \\lbrace 0, 2 \\rbrace, \\lbrace 1, 2 \\rbrace, \\lbrace 1, 3 \\rbrace$. For example, if we assign directions to each edge as $0 \\to 1$, $2 \\to 0$, $2 \\to 1$, $1 \\to 3$, then $(d_0, d_1, d_2, d_3) = (1, 2, 0, 1)$ is obtained.\nThe possible sequences $(d_0, d_1, d_2, d_3)$ are $(0, 1, 2, 1)$, $(0, 2, 1, 1)$, $(0, 2, 2, 0)$, $(0, 3, 1, 0)$, $(1, 0, 2, 1)$, $(1, 1, 1, 1)$, $(1, 1, 2, 0)$, $(1, 2, 0, 1)$, $(1, 2, 1, 0)$, $(1, 3, 0, 0)$, $(2, 0, 1, 1)$, $(2, 1, 0, 1)$, $(2, 1, 1, 0)$, $(2, 2, 0, 0)$, for a total of $14$."],["20\n00001100111010100101","261339902"]],"created_at":"2026-03-03 11:01:14"}}