MK-MO 2021 Problem 2

In the City of Islands there are $2021$ islands connected by bridges. For any given pair of islands $A$ and $B$ , one can go from island $A$ to island $B$ using the bridges. Moreover, for any four islands $A_1, A_2, A_3$ and $A_4$ : if there is a bridge from $A_i$ to $A_{i+1}$ for each $i \in \left \{ 1,2,3 \right \}$ , then there is a bridge between $A_{j}$ and $A_{k}$ for some $j,k \in \left \{ 1,2,3,4 \right \}$ with $|j-k|=2$ . Show that there is at least one island which is connected to any other island by a bridge.