
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
Given set  $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$ .Let  $a$  and  $n$  natural numbers such that  $a+2^k\in S$  for all  $k=1,2,3,...,n$ .Find the greatest value of  $n$ .

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