
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 $\{a_n\}$, the sum of the first $n$ terms is $S_n=pn^2-2n$ ($p\in \mathbb{R}, n\in \mathbb{N}^*$), and the arithmetic mean of $a_1$ and $a_5$ is $18$.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ If $a_n=2\log_2 b_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$.

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