This is the 347th most frequent Ukrainian word.
своєю
"Своєю" means "one's own" (feminine instrumental case) in English.
In this sentence, 'своєю' is the instrumental case form of the reflexive pronoun 'свій', modifying the noun 'країною' to indicate possession in the instrumental case.
Я пишаюся своєю країною.
I am proud of my country.
Here, 'своєю' takes the instrumental case agreeing with 'інтуїцією'. It indicates that the intuition belongs to Maria.
Марія завжди керується своєю інтуїцією.
Maria always follows her intuition.
In this sentence, 'своєю' agrees in case (instrumental) and gender with 'машиною', showing possession by the boys.