
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 such that  $n \geq 2$ . Let  $x_1, x_2,..., x_n$  be  $n$  distinct positive integers and  $S_i$  sum of all numbers between them except  $x_i$  for  $i=1,2,...,n$ . Let  $f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}.$ Determine maximal value of  $f(x_1,x_2,...,x_n)$ , while  $(x_1,x_2,...,x_n)$  is an element of set which consists from all  $n$ -tuples of distinct positive integers.

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