AT_agc070_b [AGC070B] Odd Namori

You are given a rooted tree $ T $ with $ N $ vertices labeled from $ 1 $ to $ N $ . The root is the vertex $ 1 $ , and the parent of vertex $ i $ $ (2 \leq i \leq N) $ is $ p_i $ $ (p_i \lt i) $ . Consider a directed graph $ G $ with $ N $ vertices labeled from $ 1 $ to $ N $ . We call $ G $ **good** if it satisfies all of the following conditions (here, let $ u \to v $ denote an edge from vertex $ u $ to vertex $ v $ ): - Every vertex in $ G $ has outdegree $ 1 $ . - $ G $ has no cycle of even length. - For every $ i $ $ (2 \leq i \leq N) $ , $ G $ does not contain the edge $ i \to p_i $ . Find the sum, modulo $ 998244353 $ , of $ 2^{(\text{number of cycles in }G)} $ over all good graphs $ G $ .

The input is given from Standard Input in the following format: > $ N $ $ p_2 $ $ p_3 $ $ \dots $ $ p_N $

Print the sum, modulo $ 998244353 $ , of $ 2^{(\text{number of cycles in }G)} $ over all good graphs $ G $ .

### Sample Explanation 1 Here are the two possible good graphs: - The graph with edges $ 1 \to 1 $ and $ 2 \to 2 $ . - The graph with edges $ 1 \to 2 $ and $ 2 \to 2 $ . The number of cycles in these graphs are $ 2 $ and $ 1 $ , respectively. Thus, the answer is $ (2^2 + 2^1) \bmod{998244353} = 6 $ . ### Sample Explanation 2 For example, the graph with edges $ 1 \to 2 $ , $ 2 \to 3 $ , and $ 3 \to 1 $ is a good graph, whose number of cycle is $ 1 $ . ### Sample Explanation 4 Remember to find the sum modulo $ 998244353 $ . ### Constraints - $ 2 \leq N \leq 10^5 $ - $ 1 \leq p_i \lt i $ - All values given in the input are integers.