
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
Consider the unordered solutions \(\left[n_{1}, \ldots, n_{N}\right]\) of the system \(n_{1} + \ldots + n_{N} = n\), \(n \geq 1, N \geq 1\), with non-negative integers \(n_{i} \geq 0\), \(i = 1, \ldots, N\). What is the number of such solutions? What is the number of such solutions if only positive solutions \(\left(n_{i} > 0\right, i = 1, \ldots, N\)) are allowed? What is the number of ordered solutions \(\left(n_{1}, \ldots, n_{N}\right)\) of the same system \(n_{1} + \ldots + n_{N} = n\), \(n \geq 1, N \geq 1\), with positive \(n_{i}\), \(i = 1, \ldots, N\)?

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