
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
Blake has six identical blue marbles, and an unlimited number of identical yellow marbles. He arranges the blue marbles and some of the yellow ones in a row such that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is a different color. For example, an arrangement like BBYYBYBYB achieves this balance. Let $m$ represent the maximum number of yellow marbles for which such an arrangement is possible, and let $N$ be the number of ways he can arrange the $m+6$ marbles to meet this condition. Calculate the remainder when $N$ is divided by 1000.

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