
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 \( p_0(x) = 0 \), \( p_1(x) = x \), \( p_{n+2}(x) = x p_{n+1}(x) + (1 - x) p_n(x) \). Find the real roots of each \( p_n(x) \).

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