
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
Consider the integral   $$ \int_{-1}^1 x^nf(x) \; dx $$   for every  $n$ -th degree polynomial  $f$  with integer coefficients.  Let  $\alpha_n$  denote the smallest positive real number that such an integral can give. Determine the limit value   $$ \lim_{n\to \infty} \frac{\log \alpha_n}n. $$  

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