Given a positive integer \(n^{}_{}\), it can be shown that every complex number of the form \(r+si^{}_{}\), where \(r^{}_{}\) and \(s^{}_{}\) are integers, can be uniquely expressed in the base \(-n+i^{}_{}\) using the integers \(1,2^{}_{},\ldots,n^2\) as digits. That is, the equation $$r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0$$ is true for a unique choice of non-negative integer \(m^{}_{}\) and digits \(a_0,a_1^{},\ldots,a_m\) chosen from the set \(\{0^{}_{},1,2,\ldots,n^2\}\), with \(a_m\ne 0^{}){}\). We write $$r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}$$ to denote the base \(-n+i^{}_{}\) expansion of \(r+si^{}_{}\). There are only finitely many integers \(k+0i^{}_{}\) that have four-digit expansions $$k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.$$ Find the sum of all such \(k^{}_{}\).

(第七届AIME1989 第14题)