LLMs can not plan. There is no LLM that can solve sudoku puzzles by executing the obvious constraint propagation algorithm with backtracking. Therefore, LLMs can neither reason nor plan. Software is not magic and the fact that a lot of people are starting to think that it is should be concerning for the folks training the next generation of software engineers.
I played asci Tic-Tac-Toe with ChatGPT using new rules that I came up with, and it was able to play using the new rules and even explained it's reasoning for the moves when I asked. I have a hard time understanding that as not including thinking and planning.
I suspect this appears to be the case because there is a ton of example pages around explaining tic-tac-toe games. My conjecture is that if you tried it with a game that wasn't so well studied as tic-tac-toe the LLM would fall flat on its face.