Pseudo-template

A string $\text{t}$ is considered a pattern of another string $\text{s}$ if the combined occurrences of $\text{t}$ within $\text{s}$ completely cover $\text{s}$. This means that every position in $\text{s}$ is covered by at least one occurrence of $\text{t}$. For example, the string $\text{aba}$ is a pattern for $\text{ababaaba}$.

A string $\text{t}$ is defined as a pseudo-pattern of a string $\text{s}$ if it meets the following criteria:

• It is a substring of $\text{s}$.

• It is a pattern of some string that contains $\text{s}$.

For example, the string $\text{abaa}$ is a pseudo-pattern of $\text{aabaabaab}$ because it appears as a substring and serves as a pattern for the string $\text{abaabaabaabaa}$.

Given a string $\text{s}$, your task is to determine the number of its pseudo-patterns and identify the shortest one.