
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
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms $S_n$ satisfying: $S_n = n^2 + 1$, and the geometric sequence $\{b_n\}$ satisfies: $b_2 = 2$, $b_5 = 16$.

$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

$(2)$ Find the sum of the first $n$ terms $T_n$ for the sequence $\{a_n b_n\}$.

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