
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 $\left\{a_{n}\right\} $ is an arithmetic sequence with the sum of its first $n$ terms being $S_{n}$, and the sequence $\left\{b_{n}\right\} $ satisfies $b_{n+2}=\frac{{b_{n+1}}^{2}}{b_{n}}\left(n\in\mathbb{N}^{*}\right)$. Also, $a_{1}=b_{1}=1$, $a_{4}+b_{4}=-20$, and $S_{4}-b_{4}=43$.

(1) Find the general term formulas for the sequences $\left\{a_{n}\right\} $ and $\left\{b_{n}\right\} $.

(2) Find the sum of the first $n$ terms, denoted as $T_{n}$, of the sequence $\left\{a_{n}\cdot b_{n}\right\} $.

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