
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 $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_5=10$, $S_{15}=240$.  
$(1)$ Find the general formula for the $n$-th term $a_n$ and the sum of the first $n$ terms $S_n$;  
$(2)$ If the sequence $\{b_n\}$ satisfies: $b_n=a_{3^n}$, find the sum of the first $n$ terms of $\{b_n\}$, denoted as $T_n$.

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