$(20$ points$)$ Given a connected undirected graph $G$ and three edges $(u_1,v_1),(u_2,v_2),(u_3,v_3)$ of

$E(G),$ design an $O(|E|)$ time algorithm to determine whether there exists a cycle in $G$ that

contains both $(u_1,v_1)$ and $(u_2,v_2),$ but does not contain $(u_3,v_3).$Given a connected undirected graph G and three edges $($u$1,$ v$1),($u$2,$ v$2),($u$3,$ v$3)$ of E$($G$),$ design an O$(|$E$|)$ time algorithm to determine whether there exists a cycle in G that contains both $($u$1,$ v$1)$ and $($u$2,$ v$2),$ but does not contain $($u$3,$ v$3).$