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With the rapid development of information technology from one side and the experience of the COVID-19 pandemic from the other side, people presently prefer to access healthcare services remotely. Telecare Medical Information System (TMIS) provides more flexible, faster, and more convenient e-healthcare services available to all people, particularly those who lack access to physicians due to their geographical restrictions. However, due to the sensitivity of medical information, preventing unauthorized access to patient data and preserving patient privacy is crucial. In this paper, we propose an authenticated key agreement scheme for TMIS to preserve the privacy of the patient's identity from all internal (even the health server and the physician) and external entities. Moreover, the physician's identity is kept secret from all external entities. Formal and informal security analysis of the proposed scheme indicates that it is secure against all attacks in the context. [ABSTRACT FROM AUTHOR]