
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 sequence of polynomials is given by the conditions:

\[ P_{0}(x) = 1, \quad P_{1}(x) = x, \quad P_{n+1}(x) = x P_{n}(x) - P_{n-1}(x), \quad n = 1, 2, \ldots \]

How many distinct real roots does the polynomial \( P_{2018}(x) \) have?

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