
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
Tara has six identical blue marbles, and an unlimited supply of identical yellow marbles. She arranges the blue marbles and some of the yellow ones in a row, ensuring that the number of times marbles whose right-hand neighbor is of the same color equals the number of times marbles whose right-hand neighbor is of a different color. Given this setup, let $m$ be the maximum number of yellow marbles for which Tara can maintain this balance, and let $N$ be the number of different arrangements she can achieve with these $m+6$ marbles altogether. Determine $N$ modulo $1000$.

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