Sentence Correction: Parallelism - A and/or/but B
This answer choice is grammatically incorrect. The construction A and B (moving on and attend) requires that A and B belong to the same part of speech. In this case, A (moving) is a non-conjugated verb, whereas B (attend) is a conjugated verb.
This answer choice is grammatically incorrect. The construction move on attending is illogical and non-idiomatic: while the verb move on indicates starting a new phase, the Verb+ing attending indicates an ongoing action. Even if this were a modifying clause separated by a comma (and then move on, attending [...]), this logical problem would remain.
There is an answer choice which better conveys the sequence of events in this sentence. Find it!
This answer choice is grammatically incorrect. It unnecessarily removes the connector and which is necessary for constructing the sentence correctly. Although both clauses in this sentence have a subject and a verb, they still must be connected by a connector.
In addition, the pronoun they in this answer choice is ambiguous as it can refer to more than one noun (students, five service academies, Beast Barracks).
This answer choice is grammatically incorrect. The verb form to attend is the object of the verb move on and should not be separated from it by a single comma:
They move on..... (to do what?).
They move on to attend a program.
Incorrect: They move on, to attend a program.
Not only does this create a fragment (to attend a four-year undergraduate program) but also it leaves the verb move on without its intended continuation.
This answer choice makes correct use of parallelism to describe to events that occur in a sequence (begin Cadet Basic Training....and then move on to attend).
summer and then move on to attend
summer, then moving on and attend
summer and then move on attending
summer, then they move on to attend
summer and then move on, to attend