
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
Let  $c$  be a nonnegative integer. Find all infinite sequences of positive integers  $\{a_{n}\}_{n\geq 1}$  satisfying the following condition: for all positive integer  $n$ , there are exactly  $a_n$  positive integers  $i$  such that  $a_i\leq a_{n+1}+c$ .

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