Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.
This is another problem from China TST 2014. I couldn't think up something. I tried to find the constant $c$ but I couldn't. I thought if I take a graph having those two cycles and a chord then we would have it, but I couldn't do it. Can someone help me? Thanks.