
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, and \( p(x) \) be a monic polynomial of degree \( n \) with real coefficients. Find the minimum possible value of \( \sum_{i=1}^{n+1} \frac{i^{2}(p(i))^{2}}{(i+1)^{2}(i+2)^{2} \ldots (i+n+1)^{2}} \).

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