
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 $f(x)$ denote the greatest power of 3 that divides $x$. For example, $f(18)=9$ and $f(27)=27$. Define for each positive integer $n$, $T_n=\sum_{k=1}^{3^n} f(3k)$. Find the smallest integer $n$ such that $T_n$ is a cube.

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