
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 an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, where the common difference $d\neq 0$, and $S_{3}+S_{5}=50$, $a_{1}$, $a_{4}$, $a_{13}$ form a geometric sequence.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $\{\frac{{b}_{n}}{{a}_{n}}\}$ be a geometric sequence with the first term being $1$ and the common ratio being $3$,<br/>① Find the sum of the first $n$ terms of the sequence $\{b_{n}\}$;<br/>② If the inequality $λ{T}_{n}-{S}_{n}+2{n}^{2}≤0$ holds for all $n\in N^{*}$, find the maximum value of the real number $\lambda$.

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