Word Chain

The White Bear and the Black Bear received a dictionary with $N$ words from Tokki. The two bears like to play word chain. In word chain, the two bears take turns saying words from the dictionary. A bear may only say a word that starts with the last letter of the word the other bear said just before. If at least one valid word exists, the bear picks one of them at random, and every valid word has the same probability of being picked. If no valid word exists, the game ends. The bear who said the last word wins. A word in the dictionary can be said many times, so the game may never end. Each time a bear says a word, Tokki decides whether the game can end. If the game cannot end no matter which words the two bears say afterward, Tokki ends the game and wins.

For each integer $i$ from $1$ to $N$, let $\alpha_i$ be the probability that the White Bear wins when the game starts with the White Bear saying the $i$-th word of the dictionary, $\beta_i$ the probability that the Black Bear wins, $\gamma_i$ the probability that Tokki wins, and $\delta_i$ the expected number of words said by the two bears until the game ends. The White Bear, the Black Bear, and Tokki compare the values computed mathematically with the values $\hat\alpha, \hat\beta, \hat\gamma, \hat\delta$ obtained by actually playing the game. Find $\alpha_i,\beta_i,\gamma_i,\delta_i$ for every integer $i$ from $1$ to $N$.

The first line contains the number of words $N$ in the dictionary. $(1\le N\le 2\times 10^5)$ From the second line onward, the $i$-th of the next $N$ lines contains the $i$-th word $w_i$ of the dictionary. $(1\le\lvert w_i\rvert\le 10;$ $i\neq j\implies w_i\neq w_j)$ Every word consists of uppercase Roman letters.

Print $N$ lines. On line $i$, print $\alpha_i,\beta_i,\gamma_i,\delta_i$, each taken modulo $998\,244\,353$ and separated by spaces. Strictly speaking, write each rational number as $\frac{P}{Q}$, where $P\ge 0$, $Q>0$, and $\gcd(P,Q)=1$ are integers. Print the unique integer $R$ with $0\le R<998\,244\,353$ that satisfies $R\times Q\equiv P\pmod{998\,244\,353}$.