A string operation breaks a string into two pieces, which costs n time units for a string of n characters. When a string is broken into many pieces, the order in which the breaks occur can affect the total amount of time used. For example, to break a 20-character string after characters 2, 8, and 10 (numbering the characters in ascending order from the left-hand end, starting from 1). If the breaks to occur in left-to-right order, then the first break costs 20 time units, the second break costs 18 time units (breaking the string from characters 3 to 20 at character 8), and the third break costs 12 time units, totaling 50 time units. If the breaks to occur in right-to-left order, however, then the first break costs 20 time units, the second break costs 10 time units, and the third break costs 8 time units, totaling 38 time units. In yet another order, the first break is at 8 (costing 20), then the 2nd break is at 2 (costing 8), and finally the third break is at 10 (costing 12), for a total cost of 40. Given a string S with n characters and an array L[1..m] containing the break points in ascending order): (1) Formulate the recursive relation of the optimal solution; (2) Design a bottom-up algorithm to calculate the lowest cost for a sequence of breaks; (3) Analyze the complexity of your algorithm; (4) Write an algorithm to print out a sequence of breaks that achieves this cost.