A semigroup at a pair of Weierstrass points on a cyclic 4-gonal curve and a bielliptic curve

We find all candidates for the Weierstrass semigroup of a pair of points on a cyclic 4-gonal curve which are total ramification points of the fixed cyclic 4-gonal map. Then we prove that such semigroups actually occur on some cyclic 4-gonal curve by constructing cyclic 4-gonal curves explicitly. Moreover, we characterize all Weierstrass semigroups of pairs of points with first nonzero nongap 4 on bielliptic curves.