
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\geq 3 $  be an integer and let  $ x_1,x_2,\ldots,x_{n-1} $  be nonnegative integers such that 

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n  x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of  $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $ .

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