TxO Math Bowl 2024 - Team Contest - Problem 30

Suppose there is a sequence of numbers S = \{a_1, a_2, a_3, \cdots, a_n\}. Define \Delta S = \{a_2 - a_1, a_3 - a_2, \cdots, a_n - a_{n-1}\}. If all elements of \Delta(\Delta(\Delta S)) = 1 and a_4 = a_9 = a_{96} = 0, find |a_1|.