A gas of hypothetical four-level atoms is at room temperature such that all atoms are initially in the ground state (
\(n=1\)). The energy levels of the atom are \(E_1 = -12.0\text{ eV}\), \(E_2 = -5.0\text{ eV}\), \(E_3 = -2.0\text{ eV}\), and \(E_4 = -1.0\text{ eV}\). Continuous white light passes through the gas to produce an absorption spectrum, while a separate electrical discharge excites a sample of these atoms up to the \(n=4\) state to produce an emission spectrum. What is the number of distinct dark absorption lines observed in the absorption spectrum, and which downward transition produces the photon with the longest wavelength in the emission spectrum?