
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 that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is ${S}_{n}={n}^{2}$, where $n \in \mathbb{N}^{*}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $A_{n}$, and it satisfies $b_{1}=a_{1}$ and $b_{n+1}=A_{n}+1$.
- $(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.
- $(2)$ Find the sum $T_{n}$ of the first $n$ terms of the sequence $\{a_{n}b_{n}\}$.
- $(3)$ Find the smallest positive integer value of $n$ that satisfies the inequality $T_{n} > 2023$.

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