Let $A_1B_1C_1D_1$ be an arbitrary convex quadrilateral. $P$ is a point inside the quadrilateral such that each angle enclosed by one edge and one ray which starts at one vertex on that edge and passes through point $P$ is acute. We recursively define points $A_k, B_k, C_k, D_k$ symmetric to $P$ with respect to lines $A_{k-1}B_{k-1}, B_{k-1}C_{k-1}, C_{k-1}D_{k-1}, D_{k-1}A_{k-1}$ respectively for $k \ge 2$. Consider the sequence of quadrilaterals $A_iB_iC_iD_i$.

i) Among the first 12 quadrilaterals, the ones that are similar to the 1997th quadrilateral are the 1st, 5th, and 9th quadrilaterals.

ii) Suppose the 1997th quadrilateral is cyclic. Among the first 12 quadrilaterals, the ones that are cyclic are the 1st, 3rd, 5th, 7th, 9th, and 11th quadrilaterals.

The answer is:
$$\begin{aligned} &\text{1. } \boxed{1, 5, 9} \\ &\text{2. } \boxed{1, 3, 5, 7, 9, 11} \end{aligned}$$