
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 an arithmetic sequence $\{a_{n}\}$, $a_{4}=4$, $S_{n}$ is the sum of the first $n$ terms of $\{a_{n}\}$, $S_{10}=55$, the sequence $\{b_{n}\}$ satisfies ${log_2}{b_1}+{log_2}{b_2}+⋯+{log_2}{b_n}=\frac{{n(n+1)}}{2}$.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ Find the sum of the first $n$ terms of the sequence $\{\left(-1\right)^{n}a_{n}b_{n}\}$ denoted by $T_{n}$.

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