To study the evolution of the student population in a university department that commenced operation in the academic year $2020-2021,$ according to the following information:

$$ It is a four$-$year undergraduate program

$$ The percentage of students progressing, repeating, or dropping out per year is approximately constant at $65-15-20\%(1$st year$),70-15-15\%(2$nd year$),85-10-5\%(3$rd year$),83-13-4\%(4$th year$)$

$$ Direct enrollment of students into the $2$nd$,3$rd$,$ and $4$th academic years is not allowed

$$ Students cannot enroll for more than $4$ years

$($a$)$ Develop the mathematical model that adequately describes the population's evolution from year to year.

$($b$)$ Record in pseudocode format the iterative algorithm for discrete$-$time simulation.