pandas dataframe column based on previous rows

I have a below dataframe

         id  action   
         ================
         10   CREATED   
         10   111
         10   222
         10   333
         10   DONE      
         10   222
         10   UPDATED   
         777  CREATED    
         10   333
         10   DONE      

I would like to create a new column "check" that would be based on data in previous rows in dataframe:

1. Find cell in action column = "DONE"
2. Search for the first CREATED or UPDATED with the same id in previous rows, before DONE. In case its CREATED then put C in case UPDATED put U.

Output:

         id  action   check
         ================
         10   CREATED   
         10   111
         10   222
         10   333
         10   DONE      C
         10   222
         10   UPDATED   
         777  CREATED    
         10   333
         10   DONE      U

I tried to use multiple if conditions but it did not work for me. Can you pls help?