
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
A sequence of polynomials \( P_{n}(x) \) (for \( n \geq 0 \) being an integer) is given recursively as \( P_{0}(x) \equiv 1 \) and \( P_{n+1}(x) = x^{17(n+1)} - P_{n}(x) \) for all \( n \geq 0 \). Find all real roots of \( P_{n}(x) \) for each \( n \geq 0 \).

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