
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  $A_o =\{1, 2\}$  and for  $n> 0, A_n$  results from  $A_{n-1}$  by adding the natural numbers to  $A_{n-1}$  which can be represented as the sum of two different numbers from  $A_{n-1}$ . Let   $a_n = |A_n |$  be the number of numbers in  $A_n$ . Determine  $a_n$  as a function of   $n$ .

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