Consider the following function which returns the number of a$$s in the input string modulo $4$:

f$($x$)=$n$\_$a $($x$)$ mod $3$

For example, the following is the result for some possible inputs:

f$($ababaabbbb$)=$n$\_$a $($ababaabbbb$)$ mod $3=4$ mod $3=1$

f$($bababab$)=$n$\_$a $($bababab$)$ mod $3=3$ mod $3=0$

Assume that we are using a unary numbering system, where $0$ is represented by a blank.

Problem: Show that the function f$($x$)$ for x in $\{$a$,$b$\}^+$ is Turing computable.

Hint: You can build this Turing machine in a modular approach, by building a machine that will take an input from the domain with a$$s and b$$s and count how many a$$s exist. The next machine will take this output as input and calculate the modulo of this integer number. You link the two machines by making the first machine$$s $($TM$1)$ final state the second machine$$s $($TM$2)$ initial state. This is basically how modular programming works, by breaking down a long program into functions that use each other$$s output successively.