
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 increasing geometric sequence $\{a_{n}\}$ with a common ratio greater than $1$, the sum of the first $6$ terms is $126$, and $4a_{2}$, $3a_{3}$, $2a_{4}$ form an arithmetic sequence. $\{b_{n}\}$ is an arithmetic sequence with $b_{2}=3$ and $S_{5}=20$.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ If $c_{n}=a_{n}b_{n}$ $(n\in N^{*})$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.

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