
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
 A set  $S$  is called *neighbouring* if it has the following two properties:
a)	 $S$  has exactly four elements
b)	for every element  $x$  of  $S$ , at least one of the numbers  $x - 1$  or  $x+1$  belongs to  $S$ .
Find the number of all *neighbouring* subsets of the set  $\{1,2,... ,n\}$ .

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