
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 positive, strictly increasing geometric sequence and $S_n$ denotes the sum of the first $n$ terms of the sequence. If $a_1 + a_3 = 5$ and $S_3 = 7$,
$(1)$ Find the general formula for the $n$-th term of the sequence $\{a_n\}$;
$(2)$ Let $b_n = (2n-1)a_n$, calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $T_n$.

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