calculus – Does this piecewise sum converge?

I am trying to study the convergence of the piecewise sum $sum_{n=1}^infty a_n$ where

$$a_n =
(1+frac{1}{43n})^{n^{43}}, & n=3p, p in mathbb{N}^{*} \
7*5^n & n=3p+1, p in mathbb{N} \
33 n, & n=3p+2, p in mathbb{N} \
(frac18)^n & n = 3p+3, p in mathbb{N}

It is obvious by the divergence test that all cases diverge, except $(frac18)^n $ which converges as a geometric sum.

But to me, it is not obvious, what $sum_{n=1}^infty a_n$ does.

Note: My intuition tells me that since there is at least one case where the sum goes to infinity (diverges) then no matter what the other cases do, the sum is going to diverge ($sum_{n=1}^infty a_n= +infty$). But I cannot prove it.

So, does the sum converge or diverge, and why?