$Z(G\ast H)=\{e\}$ and $G\ast H$ is an infinite group

Question

Let $G,H$ be a group.

Consider the following statement:

If $G$ and $H$ each have more than one element then,

$(i)$ $Z(G\ast H)=\{e\}$

$(ii)$ $G\ast H$ is an infinite group.

How can I get $(i)$ and $(ii)$?.I am seeking a suggestion. Thanks