
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  $n$  be a positive integer. We call a  $n$ -tuple  $(a_1, . . . , a_n)$  of positive integers *nice* if $\bullet$   $gcd (a_1, . . . , a_n) = 1$ , and $\bullet$   $a_i|a_{i-1} + a_{i+1}$ , for all  $i = 1, . . . , n$  (we define  $a_0 = a_n$  and  $a_{n+1} = a1$  here).
Find the maximal possible value of the sum  $a_1 +...+ a_n$  if  $(a_1, . . . , a_n)$  is a nice  $n$ -tuple.

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