Problem with 2001 Accord EX programming keyfobs

I have a 2001 Accord EX V6 coupe, which I bought used with a single
keyfob for the keyless entry system. That keyfob worked fine until I
purchased another keyfob and starting trying to add the second.

At first, when adding the new keyfob, after the programming sequence,
only the fob whose Lock button I pushed last would work (after using
the other to enter the programming mode). Either one remote or the
other would work; never both. The programming sequence (4 on-off)
seemed to go normally, with both remotes causing the locks to cycle
during the programming mode.

That was better than where I am now. After trying the above a dozen
times or so, with each showing the same result of only one fob
working, both fobs quit working entirely. Now, after going through
the programming sequence, with both fobs, or either one of the two,
the locks cycle as they should but, after exiting the vehicle and
closing the door, neither will activate any lock or trunk.

Any ideas would be appreciated.