
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 sequence $\{a_{n}\}$ is a geometric sequence with a common ratio greater than $0$, and the sequence $\{b_{n}\}$ is an arithmetic sequence, with $a_{1}=b_{1}=2$, $2a_{2}+a_{3}=a_{4}$, and $b_{3}=a_{1}+a_{2}+a_{3}$. Find:<br/>$(Ⅰ)$ The general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(Ⅱ)$ Let $c_{n}=a_{n}+b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $S_{n}$;<br/>$(Ⅲ)$ Let $d_{n}=a_{n}b_{n}$, find the sum of the first $n$ terms of the sequence $\{d_{n}\}$ denoted by $T_{n}$.

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