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