5 Each path from \(S\) to \(T\) in the network below represents a possible way of using the internet to buy a ticket for a particular event. The number on each edge represents a charge, in pounds, with a negative value representing a discount. For example, the path SAEIT represents a ticket costing \(23 + 5 - 4 - 7 = 17\) pounds.
\includegraphics[max width=\textwidth, alt={}, center]{172c5c92-4254-4593-b741-1caa83a1e833-12_1023_1330_540_350}

1. By working backwards from \(\boldsymbol { T }\) and completing the table on Figure 4, use dynamic programming to find the minimum weight of all paths from \(S\) to \(T\).
2. State the minimum cost of a ticket for the event and the paths corresponding to this minimum cost.
   (3 marks)
3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4}

   Stage	State	From	Value
   1	I	\(T\)	-7
   	\(J\)	\(T\)	-6
   	\(K\)	\(T\)	-5
   2	\(E\)	\(I\)	\(- 7 - 4 = - 11\)
   	F	I
   		\(J\)
   	G	I
   		\(J\)
   		\(K\)
   	\(H\)	\(K\)
   3

   \end{table}