> is a machine following an algorithm (even a stochastic algorithm) “thinking” or “reasoning”?

If the process is sound, yes, it can be said to be thinking or reasoning - it produced "sound thought or reasoning".

The "reasoning" is there and elsewhere the absolute (from the producing entity) output which is valid as a representation of proper reasoning.

If "A is B and B is C, then A is C" is reasoning, than what produces it in a way "reasons". If not, then a proper definition of "reasoning" must be provided, for it to be discussed.