
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
Find all positive integer  $n$  and nonnegative integer  $a_1,a_2,\dots,a_n$  satisfying: $i$  divides exactly  $a_i$  numbers among  $a_1,a_2,\dots,a_n$ , for each  $i=1,2,\dots,n$ .
( $0$  is divisible by all integers.)


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