
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
Find all pairs of  $ (a, n) $  natural numbers such that  $ \varphi (a ^ n + n) = 2 ^ n. $  
( $ \varphi (n) $  is the Euler function, that is, the number of integers from  $1$  up to  $ n $ , relative prime to  $ n $ )

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