Construct a Syntax$-$Directed Translation scheme that takes strings of a$\text{'}$s$,$ b$\text{'}$s and c$\text{'}$s as input and produces as output the number of substrings in the input string that correspond to the pattern $(b(b|)|).$ For example, the translation of the input string "abbcabcababc" is $"3"$ as shown in the following figure:

abbcabcababc

Your solution should include:

a$)$ A context$-$free grammar that generates all strings of a$\text{'}$s$,$ b$\text{'}$s and c$\text{'}$s$.$

b$)$ Semantic attributes for the grammar symbols order on the tree.

c$)$ For each production of the grammar a set of rules for evaluation of the semantic attributes.

d$)$ A justification that your solution is correct.