
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
Call a subset  $ S$  of  $ \{1,2,\dots,n\}$  *mediocre* if it has the following property: Whenever  $ a$  and  $ b$  are elements of  $ S$  whose average is an integer, that average is also an element of  $ S.$  Let  $ A(n)$  be the number of mediocre subsets of  $ \{1,2,\dots,n\}.$  [For instance, every subset of  $ \{1,2,3\}$  except  $ \{1,3\}$  is mediocre, so  $ A(3)\equal{}7.$ ] Find all positive integers  $ n$  such that  $ A(n\plus{}2)\minus{}2A(n\plus{}1)\plus{}A(n)\equal{}1.$ 

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