A 28-year-old man from Delhi is diagnosed with β-thalassemia major after presenting with severe anaemia and hepatosplenomegaly. Genetic counselling is offered to his parents. To identify the specific β-globin gene mutation in the proband and enable carrier screening in the family, the genetic laboratory receives a blood sample. The mutation is suspected to be a point mutation (not a large deletion). Which molecular technique should be performed first to detect and characterize this point mutation?