Extinction probability for a splitting amoeba

A colony starts with one amoeba. Every minute each amoeba independently either dies, stays as it is, splits into two, or splits into three, each with probability $\tfrac14$. What is the probability that the colony eventually dies out?

$\sqrt2-1\approx0.414$. Conditioning on the first minute, the extinction probability $p$ solves $p=\tfrac14(1+p+p^2+p^3)$, i.e. $(p-1)(p^2+2p-1)=0$. Since the mean offspring is $1.5>1$, extinction is the smallest root in $[0,1]$, namely $\sqrt2-1$.