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You could start with arriving at Bangalore for this city is well connected to many places in India. Spend two full days in Bangalore covering Lalbagh, Bangalore palace, Isckon temple, Tipu sultan summer palace, Vidhan soudha, planaterium, museum, Ulsoor lake, and few shopping places like MG road. From Bangalore Mysore is 4 hours away. You need a day in Mysore that from Mysore ooty is nearby. Two full days are much needed in this hill station whereby Coonoor needs half day covering. Coming back to Mysore you could cover hampi then take a bus back to Bangalore and fly to Kolkata. Each place travel takes about 5 to 6 hours so one or two days are much needed to cover each spot you have mentioned.
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