In an article published in $1965,$ Gordon Moore observed that integrated circuit $($IC$)$ technology doubled in complexity, number of transistors per IC chip, every $12$ months. $($a$)$ If complexity was $10$ in $1965,$ what would it be in $2023$ assuming Moore was correct?$($b$)$ If Moore was correct until $1990,$ what has been the doubling period on average since then? Assume complexity in $2023$ is $1010.$