
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 \( T_{L} = \sum_{n=1}^{L} \left\lfloor \frac{n^3}{9} \right\rfloor \) for positive integers \( L \). Determine all \( L \) for which \( T_{L} \) is a square number.

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