Given below are three statements and four conclusions drawn based on the statements.
Statement $1:$ Some engineers are writers.
Statement $2:$ No writer is an actor.
Statement $3:$ All actors are engineers.
Conclusion $\text{I}:$ Some writers are engineers.
Conclusion $\text{II}:$ All engineers are actors.
Conclusion $\text{III}:$ No actor is a writer.
Conclusion $\text{IV}:$ Some actors are writers.
Which one of the following options can be logically inferred?
- Only conclusion $\text{I}$ is correct
- Only conclusion $\text{II}$ and conclusion $\text{III}$ are correct
- Only conclusion $\text{I}$ and conclusion $\text{III}$ are correct
- Either conclusion $\text{III}$ or conclusion $\text{IV}$ is correct