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Gaseous ammonia has previously been demonstrated as a compelling gain medium for a quantum cascade laser pumped molecular laser (QPML), exhibiting good power efficiency but limited tunability. Here we explore the potential of the ammonia QPML to produce powerful, broadly tunable terahertz frequency lasing on rotational and pure inversion transitions. After theoretically predicting possible laser frequencies, pump thresholds, and efficiencies, we experimentally demonstrate unprecedented tunability -- from 0.762 to more than 4.5 THz -- by pumping Q- and R-branch infrared transitions with widely tunable quantum cascade lasers. We additionally demonstrate two types of multi-line lasing: simultaneous pure inversion and rotation-inversion transitions from the same pumped rotational state, and cascaded lasing involving transitions below the pumped rotational state. We report single frequency power levels as great as 0.33 mW from a low volume laser cavity., Comment: 6 pages, 4 figures, 1 tables. Supplement: 13 pages, 5 figures, 4 tables