
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
Original in Hungarian; translated with Google translate; polished by myself.

For every degree  $n$  polynomial  $f$  with integer coefficients, consider the integral   $$ \int\nolimits_{-1}^1x^nf(x)dx. $$  Let  $\alpha_n$  denote the smallest positive real number that can be the value of such an integral. Determine the limit   $$ \lim_{n\to\infty}\frac{\log\alpha_n}n. $$  

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