J. ABC

**Definitions** Let $ n \in \mathbb{Z}^+ $ be the number of strings. Let $ S = \{s_1, s_2, \dots, s_n\} $ be a set of non-empty strings over lowercase English letters. Let $ \mathcal{L} = \bigcup_{i=1}^n \{ \text{non-empty proper substrings of } s_i \} $ be the multiset of all non-empty proper substrings across all strings. For each string $ s_i $, define: - $ c_i = |\{ s_j \in S \mid s_j \text{ is a non-empty proper substring of } s_i \}| $, the number of strings in $ S $ that appear as non-empty proper substrings of $ s_i $. - $ t_i = \sum_{u \in \text{substrings}(s_i),\, u \ne s_i} \mathbf{1}_{u \in S} $, the total count of occurrences of strings in $ S $ as non-empty proper substrings of $ s_i $. **Constraints** 1. $ 1 \le n \le 10^5 $ 2. $ \sum_{i=1}^n |s_i| \le 10^6 $ 3. All characters in $ s_i $ are lowercase English letters. 4. Transition from $ s_i $ to $ s_j $ is possible iff $ s_j $ is a non-empty proper substring of $ s_i $, with probability proportional to the number of times $ s_j $ appears as a substring in $ s_i $. 5. The process stops when no non-empty proper substring of the current string belongs to $ S $. **Objective** Let $ E_i $ denote the expected number of strings visited starting from $ s_i $. Then: $$ E_i = 1 + \sum_{\substack{s_j \in S \\ s_j \text{ is a proper substring of } s_i}} \left( \frac{\text{count of } s_j \text{ in } s_i}{t_i} \cdot E_j \right), \quad \text{if } t_i > 0 $$ $$ E_i = 1, \quad \text{if } t_i = 0 $$ The answer is the average over initial choices: $$ \mathbb{E} = \frac{1}{n} \sum_{i=1}^n E_i $$ Compute $ \mathbb{E} \mod 998244353 $, expressed as $ p \cdot q^{-1} \mod 998244353 $, where $ \mathbb{E} = \frac{p}{q} $ in lowest terms.