Find the remainder when $ 12!^{14!} +1 $ is divided by $13$
I faced this problem in one of my recent exam. It is reminiscent of Wilson's theorem. So, I was convinced that $12! \equiv -1 \pmod {13} $ after this I did some test on the exponent and it seems like $12!^{n!} +1\equiv 2\pmod {13}\forall n \in \mathbb{N}$.
After I came back home I ran some more test and I noticed that if $p$ is prime then $(p-1)!^{n!} +1\equiv 2\pmod {p}\forall n \in \mathbb{N}$.
I was wondering if this result is true, if yes how to prove it? If not what is the formal way for solving the mother problem.