B$4$ For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1$dots,$t_n,$ let $f(t)$ be the corresponding real$-$valued function defined for $110$ by the following properties:

$($a$)f(t)$ is continuous for $t{t}_0,$ and is twice differentiable for all $t>i_0$ other than $t_1,$dots,$t_n$;

$($b$)f(t_0)=\frac{1}{2}$;

$($c$)\underset{t{t}_k^{+}}{lim}{f}^{\text{'}}(t)=0$ for $0kn$;

$($d$)$ For $0kn-1,$ we have $f^{\text{'}\text{'}}(t)=k+1$ when $f^{\text{'}\text{'}}(t)=n+1t>t_{n}n{t}_0,t_1$dots,$t_n{t}_k{t}_{k-1}+11knTf(t_0+T)=2023t_k,$ and $f^{\text{'}\text{'}}(t)=n+1$ when $t>t_n.$

Considering all choices $ofn$ and $t_0,t_1$dots,$t_n$ such that $t_k{t}_{k-1}+1$ for $1kn,$ what $is$ the least possible value $ofT$ for which $f(t_0+T)=2023?$