Start by understanding the relationship between the DP table and the optimal solution. How does each entry in the DP table represent a subproblem solution, and how can these be combined to form the final solution?

Consider the principle of optimality in dynamic programming. How does the DP table ensure that the solution to the larger problem is built from optimal solutions to its subproblems? Prove that the DP table correctly captures this property.

To reconstruct the optimal solution from the DP table, analyze how the choices made at each step (e.g., decisions in a knapsack or path selection in a grid) are recorded or can be inferred from the table. Provide a formal argument showing that the reconstruction process correctly identifies the sequence of choices that lead to the optimal solution.