Problem $1$: Optimal String Segmentation with Constraints

You are given a long string $\backslash ($ S $\backslash )$ and a list of forbidden substrings $\backslash ($ F $\backslash ).$ The goal is to segment the string S into the minimum number of nonempty parts such that:

$1.$ No segment contains any forbidden substring from $\backslash ($ F $\backslash ).$

$2.$ The segmentation must minimize the total number of parts.

$3.$ The difference between the size of the segments is minimum.

$4.$ Handle cases where no valid segmentation is possible, and output a message indicating failure.

$5.$ Consider strings with repetitive patterns or strings that contain every forbidden substring.

Example

$\backslash (\backslash$cdot $\backslash )$ Input: S $=$ "abcdeabc", F $=["$bc$","$de$",$ "abcd"$]$

$\backslash (\backslash$cdot $\backslash )$ Output: Segmentation into $["$a$","$bcd$",$ "eabc"$]$ is invalid, so the minimum valid segmentation is $["$abc$",$ "deabc"$].$