
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
Given a sequence $\{a_{n}\}$ where all terms are positive numbers, the sum of the first $n$ terms is $S_{n}$. It is known that $a_{1}^{3}+a_{2}^{3}+a_{3}^{3}+\ldots +a_{n}^{3}=S_{n}^{2}+2S_{n}$. Let $b_{n}=\frac{a_{n}}{2^n}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$. The smallest value of $m$ for which $T_{n} \lt m$ holds is ____.

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