
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
A $9\times 1$ board is completely covered by tiles without overlap; each tile may be $1\times 1$, $2\times 1$, or $3\times 1$ in size and lies completely on the board. Each tile is either red, blue, or green. Calculate the number of ways to tile the $9\times 1$ board such that all three colors are used at least once. Let $N$ denote this number, and give the remainder when $N$ is divided by $1000$.

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