MCR Problem of the Week, April 1, 2024 Edition

There is a pile of $N$ coins $(N \ge 3)$, exactly $1$ of which is fake. All of the real coins have the same weight, but the fake one is either slightly heavier or slightly lighter than the real ones. You have access to a weighing scale that can indicate which side is heavier, or if both sides are equally heavy. You don't have masses, so you can only weigh the coins against each other. Determine the maximum value of $N$, such that you can always find the fake coin within $3$ times of using the scale, if:

(a) you are certain that the fake coin is slightly lighter than the real coins
(b) you are not sure if the fake coin is lighter or heavier

Answer Submission Note(s)
Separate your answers with a comma.