Question 77
For a sequence of integers, determine the maximal number of distinct values that can be attained infinitely many times by this sequence if there exists a positive integer $N$ such that, for any $n \ge N$,
$$
a_n = |\{i \mid 1 \le i \lt n \text{ and } a_i + i \ge n\}|$$
$$
a_n = |\{i \mid 1 \le i \lt n \text{ and } a_i + i \ge n\}|$$