
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: $a_4=7$, $a_{10}=19$, and its sum of the first $n$ terms is $S_n$.

$(1)$ Find the general formula for the $n$-th term $a_n$ and the sum of the first $n$ terms $S_n$;

$(2)$ If the sum of the first $n$ terms of a geometric sequence $\{b_n\}$ is $T_n$, and $b_1=2$, $b_4=S_4$, find $T_n$.

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