
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
Determine all non empty sets  $C_1, C_2, C_3, \cdots $  such that each one of them has a finite number of elements, all their elements are positive integers, and they satisfy the following property: For any positive integers  $n$  and  $m$ , the number of elements in the set  $C_n$  plus the number of elements in the set  $C_m$  equals the sum of the elements in the set  $C_{m + n}$ .

*Note:* We denote  $\lvert C_n \lvert$  the number of elements in the set  $C_n$ , and  $S_k$  as the sum of the elements in the set  $C_n$  so the problem's condition is that for every  $n$  and  $m$ :
\[\lvert C_n \lvert + \lvert C_m \lvert = S_{n + m}\]
is satisfied.

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