
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 geometric sequence, with $a_2=4$ and $a_3+2$ is the arithmetic mean between $a_2$ and $a_4$.

**(Ⅰ)** Find the general term formula for the sequence $\{a_n\}$;  
**(Ⅱ)** Let $b_n=2\log_2 a_n -1$, then find the sum of the first $n$ terms of the product sequence $\{a_n b_n\}$, denoted as $T_n$.

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