
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 $k+1$ or $k+2$.
Let $T$ be the number of different towers that can be constructed. What is the remainder when $T$ is divided by 500?

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