
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  $7 \times 1$  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, blue, or green.  Let  $N$  be the number of tilings of the  $7 \times 1$  board in which all three colors are used at least once.  For example, a  $1 \times 1$  red tile followed by a  $2 \times 1$  green tile, a  $1 \times 1$  green tile, a  $2 \times 1$  blue tile, and a  $1 \times 1$  green tile is a valid tiling.  Note that if the  $2 \times 1$  blue tile is replaced by two  $1 \times 1$  blue tiles, this results in a different tiling.  Find the remainder when  $N$  is divided by  $1000$ .

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