
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 all terms are positive in the geometric sequence $\{a_n\}$, the first three terms are $a-2$, $4$, $2a$. Let the sum of the first $n$ terms be $S_n$.  
$(1)$ If $S_k=62$, find the values of $a$ and $k$;  
$(2)$ Let $b_n=(2n-1)a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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