This question tests your understanding of participles and how they are used to show the sequence of events in a sentence.

The sentence describes two actions:

1. The astronauts received their final check.
2. Then, they boarded their spacecraft.

The blank needs a word that connects these two actions, showing that the first action was completed before the second one began. We need a form of the verb "receive" that can perform this connecting function.

Let's analyze the options:

• Received: This is a simple past tense verb. If we use it, the sentence would have two main verbs ("received" and "boarded") without a conjunction to connect them, which is grammatically incorrect. Incorrect.
• To be received: This is a passive infinitive. It expresses purpose or something that is going to happen in the future, not an action that was already completed. Incorrect.
• To have received: This is a perfect infinitive. It is generally used after verbs like "seem" or "appear" (e.g., He seems to have received the check). It doesn't fit the structure of this sentence. Incorrect.
• Having received: This is a perfect participle. The perfect participle (Having + past participle) is specifically used to show that one action was completed before another action in the past. This is the perfect fit for our sentence. Correct.

Final Answer: Having received

The complete sentence is: "Having received their final check, the astronauts boarded their spacecraft." This clearly means that first they got their check, and after that was done, they boarded the craft.

Related Topic: Participles

Participles are verb forms that function as adjectives. There are two main types:

1. Present Participle (verb + -ing): Shows an action happening at the same time as the main verb.
   Example: Walking to the store, she saw a friend. (She was walking and saw at the same time).
2. Past Participle (usually verb + -ed): Often used in passive voice or perfect tenses.
   Example: Built in 1920, the house is very old. (The action of building was completed in the past).
3. Perfect Participle (Having + past participle): Emphasizes that one action was completed before another action began.
   Example: Having finished his homework, he went out to play. (First he finished, then he played).

Key Formula/Structure

The structure for showing a completed prior action is:

Having + Past Participle (Verb³), + Main Clause

This structure is used to combine two sentences where one action precedes the other.

Example: He completed the project. He submitted it to his manager.
Combined: Having completed the project, he submitted it to his manager.