
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 non empty subset  $ S$  of  $ \mathbb{N}: \equal{} \{0,1,2,\ldots\}$  such that  $ 0 \in S$  and exist two function  $ h(\cdot): S \times S \to S$  and  $ k(\cdot): S \to S$  which respect the following rules:
i)  $ k(x) \equal{} h(0,x)$  for all  $ x \in S$ 
ii)  $ k(0) \equal{} 0$ 
iii)  $ h(k(x_1),x_2) \equal{} x_1$  for all  $ x_1,x_2 \in S$ .
*(Pierfrancesco Carlucci)*

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