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Reconfigurable intelligent surfaces (RISs) are one of the most promising technological enablers of the next (6th) generation of wireless systems. In this paper, we introduce a novel use-case of the RIS technology in radio localization, which is enabling the user to estimate its own position via transmitting orthogonal frequency-division multiplexing (OFDM) pilots and processing the signal reflected from the RIS. We demonstrate that user localization in this scenario is possible by deriving Cramer-Rao lower bounds on the positioning error and devising a low-complexity position estimation algorithm. We consider random and directional RIS phase profiles and apply a specific temporal coding to them, such that the reflected signal from the RIS can be separated from the uncontrolled multipath. Finally, we assess the performance of our position estimator for an example system and show that the proposed algorithm can attain the derived bound at high signal-to-noise ratio values.