
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 $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, $2S_n=(2n+1)a_n-2n^2$ $(n\in\mathbb{N}^*)$, and a sequence $\{b_{n}\}$ satisfying $b_{1}=a_{1}$ and $nb_{n+1}=a_{n}b_{n}$. Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$. If the sum of the first $n$ terms of the sequence $\left\{\frac{{a}_{n}}{{b}_{n}}\right\}$ is denoted as $T_{n}$, and the inequality $λ+\frac{n+9}{{2}^{n}}≥4-{T}_{n}$ holds for all $n\in\mathbb{N}^{*}$, determine the range of the real number $\lambda$.

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