
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
For each even positive integer $x$, let $h(x)$ denote the greatest power of 3 that divides $x$. For example, $h(18)=9$ and $h(9)=9$. For each positive integer $n$, let $T_n=\sum_{k=1}^{3^{n-1}}h(3k)$. Find the greatest integer $n$ less than 1000 such that $T_n$ is a perfect cube.

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