Replace "Not working" part with this:
INNER JOIN (
SELECT PATIENT_ID, MIN(Date_Written) MIN_START, MAX(EXPIRES_ON) MAX_START
FROM [dbo].[PT_CASE]
GROUP BY PATIENT_ID ) AS Ptc ON [Pat].[PATIENT_ID] = [Ptc].[PATIENT_ID]
and simply add MIN_START, MAX_START to your SELECT.
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Code for TallyGenerator