A company uses drones to deliver food packages to different locations. On the first day, the drone delivers \(1\) food packages, on the second day it delivers \(2\) food packages, on the third day it delivers \(3\) food packages, and so on. The company delivers a total of \(1540\) food packages over \(n\) days.
Answer the following questions:
1. Determine the number of days \(n\) for which the total number of food packages delivered is \(1540\).
2. Find the number of food packages delivered on the last day.
3. If \(20\%\) of the total food packages records were duplicate entries, find the actual number of distinct grocery orders delivered.
4. Suppose the drone starts recording deliveries from the th day. The numbers of grocery orders delivered from the 12th day. If the total number of grocery orders delivered from the 12th day to the th day is \(2079\), obtain the number of grocery orders delivered on the last day.