Interleaved Output: Part 2 timer 20s memory 1024MB

You do not need to read the Interleaved Output: Part 1 problem to be able to solve this problem. Both Part 1 and Part 2 have the same first two paragraphs (not including this informational text). We have underlined the critical difference between the two parts.

On a distant moon of Jupiter, some developer conference events are about to happen! They are called IO (uppercase I, uppercase O), Io (uppercase I, lowercase o), iO (lowercase I, uppercase O), and io (lowercase I, lowercase O).

The best way to advertise an event is by using special computers that print the event's name one character at a time, with the output appearing on a digital display. Each such computer only knows the name of one event, and is programmed to print its event's name zero or more times. For example, a computer programmed to print IO twice prints an I, followed by an O, followed by an I, followed by an O, for a final string of IOIO.

You know that the conference organizers are using exactly one computer to advertise each event. Each printer may print its event name zero or more times. Moreover, the computers are not necessarily all programmed to print the same number of times.

The computers have all finished printing, but unfortunately, they all printed to the same display! Because the computers printed concurrently, event names in the final output string may be interleaved. You are considering the possible ways in which that string could have been produced.

For example, the string IiOioIoO could have been produced as follows:

    index:  1 2 3 4 5 6 7 8
    IO:     . . . . . . . .
    Io:     I . . . o I o .
    iO:     . i O i . . . O
    io:     . . . . . . . .
    string: I i O i o I o O
  

In this interpretation, the Io event was advertised twice, the iO event was advertised twice, and the other two events were not advertised at all.

Notice that there is no valid interpretation of this string in which the IO computer advertised its event twice. In that case, the remaining output, iioo, would have had to have come from the io computer, but that is impossible — that computer would have had to have printed i twice in a row, which is not allowed.

However, it is possible that the IO computer advertised its event once, as in the following interpretation:

    index:  1 2 3 4 5 6 7 8
    IO:     . . . . . I . O
    Io:     I . . . o . . .
    iO:     . i O . . . . .
    io:     . . . i . . o .
    string: I i O i o I o O
  

Given a final output string, what is the maximum possible number of times that the event IO could have been advertised?

It is guaranteed that the string has at least one valid interpretation. For example, oI, IOI, and IIOO are not valid inputs.