
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 sum of all positive real solutions $x$ to the equation \[2\cos2x \left(\cos2x - \cos{\left( \frac{1007\pi^2}{x} \right) } \right) = \cos4x - 1,\]where $x$ is measured in radians.

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