Problem Link
If we pick A[i] the 2nd time, it means we have a cycle.
Proof:
1st time we pick A[i], the sum before adding A[i] is x; 2nd time we pick A[i], the sum before adding A[i] is y; For this to happen x % N == y % N must hold. Otherwise we would not arrive at index i again.
x -> x + A[i]
y -> y + A[i]
(x + A[i]) % N == (y + A[i]) % N. This means we'll repeat the same set of picks in order indefinitely until we run out of operations.
By pigeon hole principle, we'll have a cycle after at most N operations.
The final answer can be constructed accordingly using the above information.