
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 positive integer. Determine the minimum of the polynomial \( f(x) = x^{2n} + 2x^{2n-1} + 3x^{2n-2} + \cdots + (2n+1-k)x^k + \cdots + 2n x + (2n+1) \) defined for real numbers.

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