
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
For a sequence  $a_1,a_2,...,a_m$  of real numbers, define the following sets 
\[A=\{a_i | 1\leq i\leq m\}\  \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\}\]
Let  $n$  be a given integer, and  $n>2$ . For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set  $A\triangle B$ , where  $A\triangle B$   $= \left(A\cup B\right) \setminus \left(A\cap B\right).$ 

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