Reordering a Geometric Sequence

A finite geometric sequence $\{a_n\}$ has $N$ terms ($N \geqslant 3$), first term $a_1 > 0$, and common ratio $q \in (0, 1)$. Its terms are rearranged into a sequence $\{b_n\}$ such that

1. For $N = 3$, list all valid sequences $\{b_n\}$. 2. Does there exist a valid $\{b_n\}$ with $b_{k-1}^2 \neq b_{k-2}b_k$ for every $3 \leqslant k \leqslant N$? Explain. 3. If a valid $\{b_n\}$ is chosen uniformly at random, find the probability that it is itself geometric.