
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Suppose that a sequence  $(a_n)_{n=1}^{\infty}$  of integers has the following property: For all  $n$  large enough (i.e.  $n \ge N$  for some  $N$  ),  $a_n$  equals the number of indices  $i$ ,  $1 \le i < n$ , such that  $a_i + i \ge n$ . Find the maximum possible number of integers which occur infinitely many times in the sequence.

Present the answer in LaTex format: \boxed{Your answer}
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