Asterisk Substrings

Consider a string $$$s$$$ of $$$n$$$ lowercase English letters. Let $$$t_i$$$ be the string obtained by replacing the $$$i$$$-th character of $$$s$$$ with an asterisk character *. For example, when $$$s = \mathtt{abc}$$$, we have $$$t_1 = \tt{*bc}$$$, $$$t_2 = \tt{a*c}$$$, and $$$t_3 = \tt{ab*}$$$.

Given a string $$$s$$$, count the number of distinct strings of lowercase English letters and asterisks that occur as a substring of at least one string in the set $$$\{s, t_1, \ldots, t_n \}$$$. The empty string should be counted.

Note that *'s are just characters and do not play any special role as in, for example, regex matching.