Let $\left\{a_n\right\}$  be the sequence of real numbers defined by $a_1=t$  and $a_{n+1}=4a_n(1-a_n),n\ge 1$.
If  $a+P^b$ is the total number of distinct values of t such that we have  $a_{2024}$= 0, where P is prime number and a is the least possible natural number to be fit in the given form, what is the value of

 a + b