If I have an automatic sequence, like the thue-morse sequence:
$0\rightarrow \{0,1\}$ and $1\rightarrow \{1,0\}$ initialized with $0$
And another sequence:
$1\rightarrow \{1,0,1\}$ and $0\rightarrow \{0,0,0\}$ initialized with $1$.
How can I figure out when both sequences have a $1$ at a given position?