Product envelope for an arbitrary finite segment list

Let $T:\mathbb N\to\mathbb N$ be the shortcut Collatz map: $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. Write $T^k$ for its $k$-fold iterate, with $T^0(n)=n$. Let $S$ be a finite list of segments $(k_i,A_i,D_i)$ of natural numbers, with $D_i>0$. Starting from $n$, define successive endpoints by applying $T^{k_i}$. Assume each segment satisfies $D_i(\text{endpoint}+3)\le A_i(\text{start}+3)$. Write $K=\sum_i k_i$, $A=\prod_i A_i$, and $D=\prod_i D_i$, with empty products 1.

Under the segment assumptions,

The list can have any finite length, including zero. All segment inequalities remain explicit premises.