Let $G(V,E)$ be an undirected graph, potentially with loops and multi-edges. Assume the following two properties hold:
- $\forall a \in V, \exists f \in \operatorname{Aut}(G), f(a) \ne a$
- $\forall f\in \operatorname{Aut}(G), \exists a \in V, f(a) = a $
Prove: $10 \le |V| $