
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
The polynomials  $P_{n}(x)$  are defined by  $P_{0}(x)=0,P_{1}(x)=x$  and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number  $n\geq 1$ , find all real numbers  $x$  satisfying the equation  $P_{n}(x)=0$ .

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