
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
Michael revisits his quadratic roots question but now considers a quadratic \[x^2 - sx + p\] with roots $r_1$ and $-r_1$. He is curious about the sum of the squares of the roots for any power up to 2005, i.e., \[r_1^2 + (-r_1)^2 = r_1^4 + (-r_1)^4 = \cdots = r_1^{2006} + (-r_1)^2006.\] Determine the maximum value of \[\dfrac{1}{r_1^{2006}} + \dfrac{1}{(-r_1)^{2006}}.\]

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