
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 magnitude of the product of all complex numbers  $c$  such that the recurrence defined by  $x_1 = 1$ ,  $x_2 = c^2 - 4c + 7$ , and  $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$  also satisfies  $x_{1006} = 2011$ .

*Author: Alex Zhu*

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