
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 each positive integer  $n\ge2$ , find a polynomial  $P_n(x)$  with rational coefficients such that  $\displaystyle P_n(\sqrt[n]2)=\frac1{1+\sqrt[n]2}$ . (Note that  $\sqrt[n]2$  denotes the positive  $n^\text{th}$  root of  $2$ .)

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