
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 of $a_{2}$ and $a_{4}$.
(Ⅰ) Find the general formula for the sequence $\{a_{n}\}$;
(Ⅱ) Let $b_{n} = 2\log_{2}a_{n} - 1$, find the sum of the first $n$ terms of the sequence $\{a_{n}b_{n}\}$, represented as $T_{n}$.

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