
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
*Version 1*. Let  $n$  be a positive integer, and set  $N=2^{n}$ . Determine the smallest real number  $a_{n}$  such that, for all real  $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
*Version 2*. For every positive integer  $N$ , determine the smallest real number  $b_{N}$  such that, for all real  $x$ ,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]

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