Prove that every undirected connected graph with $|V | > 2$ results in a connected graph if two vertices removed.. How do I go about proving this?

Prove that every undirected connected graph with $|V | &gt; 2$ has at least two vertices such that if one or both are removed (along with their incident edges) the resulting graph is still connected, and describe an efficient algorithm to find two such vertices.
Hint: Find the connected graph's spanning tree (using an algorithm like breadth-first search). Consider the endpoints of a longest path in this spanning tree.