
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 odd positive integer \( y \), let \( h(y) \) denote the greatest power of 3 that divides \( y \). For example, \( h(9) = 9 \) and \( h(27) = 27 \). For each positive integer \( m \), let \( T_m = \sum_{k=1}^{3^{m-1}} h(3k-1) \). 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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