
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
Ed has six identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row such that the number of marbles whose right hand neighbor is the same color is equal to the number of marbles whose right hand neighbor is a different color. If $m$ is the maximum number of red marbles for which such an arrangement is possible, determine the number of ways he can arrange these $m+6$ marbles to satisfy the requirement and find the remainder when this number is divided by 1000.

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