
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 collection of 9 cubes consists of one cube with edge-length \(k\) for each integer \(k, 1 \le k \le 9\). A tower is to be built using all 9 cubes according to the rules:
- Any cube may be the bottom cube in the tower.
- The cube immediately on top of a cube with edge-length \(k\) must have edge-length at most \(k+3\).
Let \(T\) be the number of different towers that can be constructed. What is the remainder when \(T\) is divided by 1000?

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