
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
In a geometric sequence $\{a_n\}$ with a positive common ratio, $a_3 - a_1 = \frac{16}{27}$ and $a_2 = -\frac{2}{9}$. Another sequence $\{b_n\}$, where $b_n > 0$, has its first $n$ terms' sum $S_n$ satisfying $S_n - S_{n-1} = \sqrt{S_n} + \sqrt{S_{n-1}}\ (n \geq 2)$, and $S_{10} = 100$.

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

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

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