
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
Find the maximum number of real roots to a polynomial of the form
\[ ax^n + x^{n-1} + \dots + x + 1 = 0, \]
where \( n \) is a positive even integer, and \( a \) is a real number not equal to 0.

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