
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 the sequence $\{a_n\}$, where $a_1=1$ and $a_{n+1}=3a_n+1$ for $n\in\mathbb{N}_+$, and let $b_n= \frac{n}{(3^n-1)\cdot2^{n-2}}\cdot a_n$, $T_n$ is the sum of the first $n$ terms of $\{b_n\}$. If the inequality $2^n\lambda < 2^{n-1}T_n+n$ holds for all $n\in\mathbb{N}_+$, then the range of the real number $\lambda$ is ______.

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