
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
Let  $m$  be the maximum possible value of  $x^{16} + \frac{1}{x^{16}}$ , where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when  $m$  is divided by  $2007$ .

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