How to choose the first action in a Monte Carlo Tree Search?

I'm working on reimplementing the MuZero paper. In the description of the MCTS (page 12), they indicate that a new node with associated state $$s$$ is to be initialized with $$Q(s,a) = 0$$, $$N(s,a) = 0$$ and $$P(s,a) = p_a$$. From this, I understand that the root node with state $$s_0$$ will have edges with zero visits each, zero value and policy evaluated on $$s_0$$ by the prediction network.

So far so good. Then they explain how actions are selected, according to the equation (also on page 12):

But for the very first action (from the root node) this will give a vector of zeros as argument to the argmax: $$Q(s_0,a) = 0$$ and $$\sum_bN(s_0,b)=0$$, so even though $$P(s_0,a)$$ is not zero, it will be multiplied by a zero weight.

Surely there is a mistake somewhere? Or is it that the very first action is uniformly random?

I can't really think of any reason why this would be better than playing according to $$P(s, a)$$ in the very first iteration though. Maybe it creates a tiny little bit more variety in the tree search results you get? I suppose all the stochasticity that is normally inherent to a vanilla MCTS is no longer present in MuZero (or AlphaZero), because they always run for exactly the same number of iterations, and don't have any sort of random rollouts anymore; this would at least introduce a tiny bit of variation.