
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
Alice has four identical green marbles, three identical red marbles, and a large supply of identical blue marbles. She arranges green, red, and some blue marbles in a row and finds that for this to be balanced, the number of blue marbles must make the pairs of neighboring marbles with the same color equal to the number of pairs of neighboring marbles with different colors. Let $m$ be the maximum number of blue marbles Alice can use for such an arrangement, and let $N$ be the number of ways she can arrange the $4+3+m$ marbles to satisfy this requirement. Calculate $N$ and find the remainder when $N$ is divided by $1000$.

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