
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 blue marbles and an unlimited number of identical yellow marbles. He arranges the blue marbles and some of the yellow ones in a line such that the number of marbles with a neighbor of the same color is equal to the number with a neighbor of a different color. Let \( m \) be the maximum number of yellow marbles for which this arrangement is possible, and let \( N \) be the number of ways he can arrange the \( m+6 \) marbles to satisfy this condition. Calculate the remainder when \( N \) is divided by \( 1000 \).

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