
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
Harry has seven identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some red ones in a circle such that the number of marbles whose neighbor (either left or right in the circle) is the same color as themselves is equal to the number of marbles with neighbors of a different color. An example of such an arrangement could be GRGRRGRGRGG. Let $m'$ be the maximum number of red marbles for which such an arrangement is possible, and let $N'$ be the number of ways he can arrange the $m' + 7$ marbles to satisfy this requirement. Find the remainder when $N'$ is divided by $1000$.

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