Compare and Connect

There is a multiset $S$ containing $N + M$ strings, $N$ of which are "$<$" and $M$ of which are "$>$". For two strings $s$ and $t$, $f(s, t)$ is defined as: - If $s$ is smaller than $t$ in lexicographical order, $f(s, t) = s +$ "$<$" $+ t$; - If $s$ is equal to $t$ in lexicographical order, $f(s, t) = s +$ "$=$" $+ t$; - If $s$ is larger than $t$ in lexicographical order, $f(s, t) = s +$ "$>$"