
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 increasing geometric sequence $\{a_n\}$ that satisfies: $a_2 \cdot a_3 = 8$ and $a_1 + a_4 = 9$  
$(1)$ Find the general formula for the sequence $\{a_n\}$.  
$(2)$ Let the sequence $\{b_n\}$ be defined as $b_n = 2(2n-1)a_n (n \in \mathbb{N}^+)$, 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}
user