Let $S[4]$ be a binary array with elements of $S$ are taken uniformly and independently from $\{0,1\}$. Also take $k$ uniformly from $\{0,1\}$. Take $i=1$. Now run the following process:
- Take $a,b$ uniformly and independently from $[1,4]$
- Change $k=(k+S[a]) \bmod 2$
- $S[b]=(k+S[i])\bmod 2$
- $i=(i+1) \bmod 4$
Run the whole process (1,2,3,4) until all elements of S are filled with 0. What is expected number of runs for this event?