Quick questions about a document example for Components and Data Structures

NDSolve can be broken into several basic steps for advanced usage according to Components and Data Structures. In the documentation, a simple example is given as follows.

1. Creating an NDSolve`StateData object, which includes the info needed to solve the equation:

    state = First[NDSolve`ProcessEquations[{x''[t] + (1 + 4 UnitStep[Sin[t]]) x[t] == 0, x[0] == 1, x'[0] == 0}, x, t, Method -> "ExplicitRungeKutta"]]
   
    (* NDSolve`StateData[<0.>] *)
   

2. Integrating the equation up to t = 10 Pi with the time interval specified by a list of intermediate times

    NDSolve`Iterate[state, Pi Range[10]]
   

3. Generating the current solution and associated derivatives in the forward directon

    sol = NDSolve`ProcessSolutions[state, "Forward"]
   
    (* {x[31.4159] -> 0.843755, Derivative[1][x][31.4159] -> -1.20016, (x'')[31.4159] -> -0.843755, (x'')[31.4159] -> (x'')[31.4159] -> -1} *)
   

I have two questions with regard to the step 2 and 3, respectively.

(1) The document states that NDSolve`Iterate allows you to specify intermediate times at which to stop. This can be useful to avoid discontinuities. It sounds great but I don’t understand what does it mean by at the intermediate times to stop. Does it mean that by specifying intermediate times we can use NDSolve`ProcessSolutions to save all the intermediate solutions otherwise we only have a solution for the final time? Can anyone give an example to demonstrate how to avoid discontinuities in solving a differential equation with NDSolve`Iterate[state, list_of_times]?

(2) I can understand that NDSolve`ProcessSolutions gives the solution and associated derivatives in the step 3, however, I do not understand what is the last element in its output list, i.e., (x'')[31.4159] -> (x'')[31.4159] -> -1?