V+자 (Suggestion) is used to make informal suggestions to do something together. The more formal version of this expression is V+(으)ㅂ시다 (Suggestion).
V+자 (Simultaneous Action) indicates that the second action happens as or when the first action is occurring. Unlike V+자마자 (Immediately after), which emphasizes that the second action happens immediately after the first, V-자 in this usage conveys that the two actions are closely connected in time, often overlapping.
Note: This usage of V+자 is more common in written or formal contexts and is less frequently used in spoken Korean. In conversational Korean, V+(으)니까 (Reason or cause) or V+아/어서 (Reason or cause) might be used instead.