
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 positive integer and let  $P(x)$  be a monic polynomial of degree  $n$  with real coefficients. Also let  $Q(x)=(x+1)^2(x+2)^2\dots (x+n+1)^2$ . Consider the minimum possible value  $m_n$  of  $\displaystyle\sum_{i=1}^{n+1} \dfrac{i^2P(i^2)^2}{Q(i)}$ . Then there exist positive constants  $a,b,c$  such that, as  $n$  approaches infinity, the ratio between  $m_n$  and  $a^{2n} n^{2n+b} c$  approaches  $1$ . Compute  $\lfloor 2019 abc^2\rfloor$ .

*Proposed by Vincent Huang*

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