
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
Let \( N \) be a given natural number. Express \( N \) as the sum of \( k \) natural numbers \( x_{1}, x_{2}, \cdots, x_{k} \). If \( N = kt + r \) (where \( t \) and \( r \) are non-negative integers), \( 0 \leq r < k \) and \( 1 < k \leq N \), then the maximum value of the product \( x_{1} x_{2} \cdots x_{k} \) is \( t^{k-r} (t+1)^{r} \).

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