
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 number of real solutions to
\[(x^{2010} + 1)(x^{2008} + x^{2006} + x^{2004} + \dots + x^2 + 1) = 2010x^{2009}.\]

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