
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
Let $k$ be a positive integer multiple of 3. Bernardo and Silvia play a game with numbers as follows: Bernardo writes the smallest integer with $k$ digits which is also a perfect cube, rather than a square. Every time Bernardo writes a number, Silvia erases the last $k-1$ digits of it. Bernardo then writes the next perfect cube, Silvia erases the last $k-1$ digits, and this process continues until the last two numbers that remain on the board differ by at least 3, rather than 2. Let $h(k)$ be the smallest positive integer not written on the board. Compute the sum of the digits of $h(3) + h(6) + h(9) + \dots + h(999)$.
A) 654
B) 655
C) 666
D) 677
E) 688

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