Start by recalling the formal definition of NP-completeness and the role of polynomial-time reductions in proving NP-completeness.

Construct a reduction from 3SAT to the Hamiltonian Cycle problem by designing a graph where a Hamiltonian cycle exists if and only if the 3SAT formula is satisfiable, ensuring each step of the reduction is polynomial-time verifiable.

Prove the correctness of the reduction by demonstrating that any satisfying assignment of the 3SAT formula corresponds to a Hamiltonian cycle in the constructed graph and vice versa, while also verifying the polynomial-time complexity of the transformation.