What does an engine do if it has found the shortest mate? For example, take this famous position from Kramnik vs. Deep Fritz, 2005:
[FEN &quot;5N1k/6p1/7p/4P3/pp2Q3/4q3/1P4PP/2b4K w - - 0 1&quot;]

Obviously White wins with Qh7#, and equally obviously, there are no possible better moves (since it's impossible to have a mate in 0).
If I let an engine run on this position, what does it do after finding the mate in 1? There's presumably nothing else to do since it has already found the best move. Does it just exit, then?
I tried running Stockfish 13 on this position, and it reaches depth 99/99 instantly, but I can't tell what it's doing afterwards.
It does what it was programmed to do.  During any Alpha-Beta pruning, the computer would stop searching and just do nothing.
In another game, like an RTS, the computer would constantly go through a todo list.
What should/can I build?
What should/can I produce?
How do I defend/attack?
...
Since a chess AI has only one objective, find the best move, there's no point in searching once it's objective if found.