
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
Let  $n$  be an even positive integer. Let  $p$  be a monic, real polynomial of degree  $2 n$ ; that is to say,  $p(x)=$   $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$  for some real coefficients  $a_0, \ldots, a_{2 n-1}$ . Suppose that  $p(1 / k)=k^2$  for all integers  $k$  such that  $1 \leq|k| \leq n$ . Find all other real numbers  $x$  for which  $p(1 / x)=x^2$ .

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