
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
In coordinate space, a set of points $E$ is given, consisting of all points whose three coordinates are integers between 0 and 1982. Each point in $E$ is colored either red or blue. The coloring is such that, for every rectangular box whose vertices are in $E$ and edges are parallel to the coordinate axes, the number of red vertices is divisible by 4. How many different colorings are there?

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