
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 $\{a_n\}$ satisfying: $a_{n+1} > a_{n}$ $(n\in{N}^*)$, $a_{1}=1$, the first three terms of the sequence, after adding $1$, $1$, $3$ respectively, form a geometric sequence, and $a_{n}+2\log_{2}{b_{n}}=-1$.

$(1)$ Find the general formula for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.

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

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