
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
In the arithmetic sequence $\{a_{n}\}$, it is known that $a_{1}+a_{3}=10$, and $a_{4}=9$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and the sum of the first $n$ terms of the sequence $\left\{{\frac{1}{{{S_n}}}}\right\}$ be $T_{n}$. If for all real numbers $n\in N^{*}$, it holds that ${T_n}＜3{λ^2}+\frac{9}{4}λ$, determine the range of real number $\lambda$.

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