You are given a string of $n$ characters $s[1$dotsn$],$ which you believe to be a corrupted text document in which all punctuation has vanished $($so that it looks something like "itwasthebestoftimes..."$).$ You wish to reconstruct the document using a dictionary$]$ which is available in the form of a Boolean function $)$: for any string $w,$

dict$(w)=\{\begin{array}{ccc}& \text{t}\text{r}\text{u}\text{e}ifwis\text{a}\text{v}\text{a}\text{l}\text{i}\text{d}\text{w}\text{o}\text{r}\text{d}\text{,}& \\ & \text{f}\text{a}\text{l}\text{s}\text{e}\text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

$($a$)$ Give a dynamic programming algorithm that determines whether the string $.$