A bead is constrained to move along a long, massless, frictionless horizontal rod parallel to the \( x \)-axis. The rod itself is moving vertically upward along the \( z \)-direction against gravity with a constant speed, starting from \( z = 0 \) at \( t = 0 \), and remains horizontal. The conjugate momenta are denoted by \( p_x, p_y, p_z \) and the Hamiltonian by \( H \). Which of the following option(s) is/are correct?