If my Familiar has the Spellcasting ability, is it limited to spells with 2 actions or less?
The ability make it sound like the only limitation is that it is 5 levels lower than your highest-level spell slot.
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You have to spend all the actions of an activity at once to gain its effects. In an encounter, this means you must complete it during your turn.
Because the familiar has the Minion trait, it can't devote 3 actions to casting a spell in encounter mode, since it can at most devote 2 actions to the spell.
There are no such restrictions if you aren't in an encounter. During exploration, a familiar could definitely cast a 3-action spell or one with a longer cast time, such as 1 minute, following the regular rules of the Cast a Spell activity.