19403번 - Cyclic Shifts 다국어

Let $s = s_1 s_2 \ldots s_k$ be a string of length $k$. For any integer $i$ between 0 and $k-1$, inclusive, we define the $i$-th cyclic shift of $s$ as the string $s_{i+1}s_{i+2} \ldots s_k s_1 \ldots s_i$. For example, the 4-th cyclic shift of "wellplayed" is "playedwell", while the 0-th cyclic shift of "metro" is "metro" itself.

Let's define a function $f(s)$ which, for a string of length $k$, is equal to $i$ such that the $i$-th cyclic shift of $s$ is the lexicographically smallest among all its cyclic shifts. If there are several such $i$'s, then $f(s)$ is equal to the smallest of them. For example, $f($"acabbac"$) = 2$, while $f($"cabcab"$) = 1$.

Let's define a function $g(s)$ which, for a string of length $n$, is equal to the sum of $f(s_1 s_2...s_k) \cdot 8753^{k-1}$ over all $k$ between $1$ and $n$, inclusive.

For each given string $s$, find the value of $g(s)$ modulo $10^9 + 123$.

The first line of the input contains a single integer $t$ ($1 \le t \le 10^4$) --- the number of test cases.

Each of the next $t$ lines contains a non-empty string consisting of lowercase English letters.

The total length of the input strings doesn't exceed $10^6$.

For each string $s$ in order of input, output a single integer --- the value of $g(s)$ modulo $10^9 + 123$.

In the first example test case, $f($"a"$) = 0$, $f($"aa"$) = 0$, and $f($"aab"$) = 0$. Therefore, $g($"aab"$) = 0$.

Here is the list of values of $f(s_1s_2 \ldots s_k)$ for the second example test case:

• $f($"a"$) = 0$;
• $f($"ac"$) = 0$;
• $f($"aca"$) = 2$;
• $f($"acab"$) = 2$;
• $f($"acabb"$) = 2$;
• $f($"acabba"$) = 5$;
• $f($"acabbac"$) = 2$.

Thus, $g($"acabbac"$) = 2 \cdot 8753^2 + 2 \cdot 8753^3 + 2 \cdot 8753^4 + 5 \cdot 8753^5 + 2 \cdot 8753^6 = 899695598935764095704157$, which is equal to $38098220$ modulo $10^9 + 123$.