
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 $\{a_n\}$, the product of its first three terms is $8$, and adding $1$, $2$, $2$ to these three terms respectively turns them into an arithmetic sequence.

$(1)$ Find the general formula for the geometric sequence $\{a_n\}$.

$(2)$ Let $b_n = a_n + 2n$, 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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