
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  $n\ge 3$ . Suppose  $a_1, a_2, ... , a_n$  are  $n$   distinct in pairs real numbers.
In terms of  $n$ , find the smallest possible number of different assumed values by the following  $n$  numbers:  $$ a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1 $$  

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