
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
For positive integers  $i = 2, 3, \ldots, 2020$ , let \[
a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}.
\]Let  $x_2$ ,  $\ldots$ ,  $x_{2020}$  be positive reals such that  $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$ . Let  $S$  be the maximum possible value of \[
\sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{-2.25} x_i)
\] and let  $m$  be the smallest positive integer such that  $S^m$  is rational. When  $S^m$  is written as a fraction in lowest terms, let its denominator be  $p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$  for prime numbers  $p_1 < \cdots < p_k$  and positive integers  $\alpha_i$ . Compute  $p_1\alpha_1+p_2\alpha_2 + \cdots + p_k\alpha_k$ .

*Proposed by Edward Wan and Brandon Wang*

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