A curious species of amoeba undergoes the following evolution rule - at each time step, with probability $p$ it splits into two copies of itself of size $\varepsilon S$, where $S$ is the original size of the amoeba, and $0 < \varepsilon < 1$ is a fixed number. With probability $q = 1-p$, it doubles in size instead.

Start with one amoeba of size $1$. Denoting by $S^{\max}_t$ the size of the largest amoeba at time $t$, is it true that we either have

$$\lim_{t \to \infty} S^{\max}_t = \infty$$

almost surely, or

$$\lim_{t \to \infty} S^{\max}_t = 0$$

almost surely, depending on $\varepsilon, p$? If so, which parameters lead to which regimes?