
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 natural numbers  $ n $  for which there exists two natural numbers  $ a,b $  such that  $$  n=S(a)=S(b)=S(a+b) ,  $$  where  $ S(k) $  denotes the sum of the digits of  $ k $  in base  $ 10, $  for any natural number  $ k. $ *Vasile Zidaru* and *Mircea Lascu*

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