$1.(10+20+10=40$ points$)$ Consider the following infinite set A over $\backslash$Sigma $=\{$Z$,$ B$,$ o$,$ i$,$ n$,$ k$,$ s$\}.$

Set A consists of triplets.

The first member of each triplet is the word $$Zoinks$$ with $1$ one or more $$o$$s$.$

The second member is the word $$Boinks$$ with the same number of $$o$$s as the first member.

The third member is the number of $$o$$s$.$

A $=\{[$Zoinks$,$ Boinks, $1],[$Zooinks$,$ Booinks, $2],[$Zoooinks$,$ Boooinks, $3],...\}$

Give a recursive definition of the set A$.$

Write all three components of the recursive definition including the closure.

$($ Basis: $10$ points, Recursive Step: $20$ points, Closure: $10$ points $)$

You will not be asked to write the closure in the exams, it will be preprinted.