
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 positive integer $x$, let $h(x)$ denote the greatest power of 3 that divides $x$. For example, $h(18)=9$ and $h(27)=27$. For each positive integer $m$, let $T_m=\sum_{k=1}^{3^{m-1}}h(3k)$. Find the greatest integer $m$ less than 500 such that $T_m$ is a perfect cube.

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