
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
The elements of the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ can be separated into the sets $\{3,9,10,11\}$ and $\{1,2,4,5,6,7,8\}$ so that each has the same sum of elements equal to 33.

(a) Describe a way to separate the elements of the set $\{1,2,3,4,5,6,7,8\}$ into three subsets such that the sum of the elements in each subset is the same.
(b) Explain why it is not possible to separate the numbers of the set $\{1,2,3,4,5,6,7,8,9,10\}$ into two subsets with the same sum.

(c) For each positive integer $n \geq 2$, determine the smallest positive integer $N$ such that the set $\{1,2, \ldots, N\}$ can be separated into exactly $n$ subsets with the same sum.

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