Compute modulo of prime power tower

Question

I have a prime number $p$, and I need to compute $p$ ↑↑ $k$ mod $m$.

here p ↑↑ k can be written as p ^ (p ^ (p ^ (p ... k times)))

for example -

$p = 5, k = 3, m = 3$

5 ↑↑ 3 = 5 ^ (5 ^ 5) % 3 = 2

Here $p$ is prime number, $2 <= p <= 10 ^ 8$, and $p - 2 <= m <= 10 ^ 8$.