Partition the set of naturals from $1$ to $16$ into two subsets satisfying certain conditions

Question

I want to partition the set $S=(1,2,3,...,16)$ into two subsets $A=(a_1,a_2,...,a_8) $ and $B=(b_1,b_2,...,b_8)$ such that:$$[1]\;\;\;\sum_{i=1}^{8}a_i=\sum_{i=1}^{8}b_i$$ And:$$[2]\;\;\;\sum_{i=1}^{8}a_i^2=\sum_{i=1}^{8}b_i^2$$ And:$$[3]\;\;\;\sum_{i=1}^{8}a_i^3=\sum_{i=1}^{8}b_i^3$$ I tried some configurations, I found that $A=(1,4,6,7,9,12,14,15)$ and $B=(2,3,5,8,10,11,13,16)$ satisfy $[1]$ and $[2]$ but not $[3]$, I figured out that the sum in $[3]$ should be $9248$ but I'm kinda stuck, I don't know what else I could do other than painstakingly compute other configurations...

Answer 1

How about $A=\{1,4,6,7,10,11,13,16\}$?