Alice-starting exact pointer jumping

There exists a real constant $C>0$ such that, for every pair of natural numbers $k,n$ with $k>0$ and $n\ge2$, and every starting index $v_1\in\{0,\ldots,n-1\}$, there exist a natural number $c$ and a public randomized classical protocol $P$ with inputs $a,b:\{0,\ldots,n-1\}\to\{0,\ldots,n-1\}$ and output in the disjoint union of two copies of $\{0,\ldots,n-1\}$, satisfying the following conditions. The protocol consists of finitely many deterministic communication trees with nonnegative real weights summing to one. Each internal node sends one Boolean determined solely by the sending party’s input and follows the corresponding branch; a leaf outputs a function of either Alice’s input or Bob’s input. Every tree of strictly positive weight has communication cost at most $c$, where cost is the maximum number of internal nodes on a path, and has at most $k$ rounds starting with Alice: an output-only tree has zero rounds, while a nonempty path counts its consecutive speaker blocks, with an additional initial round if its first speaker is Bob. For every $a,b$, the total weight of trees whose output is $\operatorname{succ}_{a,b}^{\,k+1}(\operatorname{inl}v_1)$ is at least $1$, where $\operatorname{succ}_{a,b}(\operatorname{inl}v)=\operatorname{inr}(a(v))$ and $\operatorname{succ}_{a,b}(\operatorname{inr}v)=\operatorname{inl}(b(v))$. Finally, $c\le Ck\log_2 n$. Trees of weight zero are not subject to the cost and round bounds.