
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
Given an integer  ${n>1}$ , denote by  $P_{n}$  the product of all positive integers  $x$  less than  $n$  and such that  $n$  divides  ${x^2-1}$ . For each  ${n>1}$ , find the remainder of  $P_{n}$  on division by  $n$ .

*Proposed by John Murray, Ireland*

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