
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 a monotonically increasing arithmetic sequence with the first term $a_1=3$, and the sum of the first $n$ terms is $S_n$. The sequence $\left\{ b_n \right\}$ is a geometric sequence with the first term $b_1=1$, and $a_2 \cdot b_2=12$, $S_3+b_2=20$.

$(1)$ Find the general formula for sequences $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$;

$(2)$ Let $c_n=a_n \cdot b_n$, find the sum of the first $n$ terms of the sequence $\left\{ c_n \right\}$, denoted as $T_n$;

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