Bound on growth of a sequence

Question

I have a sequence $a_n$ which satisfies the bound \begin{equation*} \tag{1} a_n\ge \frac{a_{n-1}}{2}+c_1\frac{a_{n-2}}{n^{1/4}}-c_2\frac{a_{n-1}^{1/2} a_{n-2}^{1/2}}{n^{1/8}}. \end{equation*} Here $c_1,c_2$ are fixed and (strictly) positive numbers, $a_n>0$ for every $n$. I would like to deduce from this a lower bound for $a_n$ in terms of $a_{n-1}$, asymptotically as $n\to\infty$. From rough scaling arguments it seems to me that (1) should be compatible with a behavior of the type \begin{equation*} a_n = c \frac{a_{n-1}}{n^{1/4}} \end{equation*} for some $c>0$ fixed. Is it possible to prove anything in this direction? Not as an equality of course, but maybe at least something like \begin{equation*} a_n \ge c \frac{a_{n-1}}{n^\alpha} \end{equation*} for $\alpha$ reasonably close to $1/4$.