
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
An $8\times 1$ board is completely covered by $m\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, yellow, or blue. Let $N$ be the number of tilings of the $8\times 1$ board in which all three colors are used at least once. For example, a $1\times 1$ red tile followed by a $3\times 1$ yellow tile, a $2\times 1$ yellow tile, a $1\times 1$ blue tile, and a $1\times 1$ yellow tile. Calculate $N$ modulo $1000$.

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