
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 an given positive integer, the set  $S=\{1,2,\cdots,n\}$ .For any nonempty set  $A$  and  $B$ , find the minimum of  $|A\Delta S|+|B\Delta S|+|C\Delta S|,$  where  $C=\{a+b|a\in A,b\in B\}, X\Delta Y=X\cup Y-X\cap Y.$ 

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