The table shows the distances, in miles, along the direct roads between six villages, \(A\) to \(F\). A dash (\(−\)) indicates that there is no direct road linking the villages.

\((\text{i})\) On the table in the insert, use Prim’s algorithm to find a minimum spanning tree. Start by crossing out row \(A\). Show which entries in the table are chosen and indicate the order in which the rows are deleted. Draw your minimum spanning tree and state its total weight.\((\text{ii})\) By deleting vertex \(B\) and the arcs joined to vertex \(B\), calculate a lower bound for the length of the shortest cycle through all the vertices.

\((\text{iii})\) Apply the nearest neighbour method to the table above, starting from \(F\), to find a cycle that passes through every vertex and use this to write down an upper bound for the length of the shortest cycle through all the vertices.

**Input the upper bound, in miles, as your answer.**

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