
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
Suppose that $\{a\_n\}$ is a geometric sequence with all positive terms, and $\{b\_n\}$ is an arithmetic sequence, with $a\_1=b\_1=1$, $a\_3+b\_5=13$, and $a\_5+b\_3=21$.

(I) Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$.

(II) Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. Find the sum of the first $n$ terms of the sequence $\{S\_n \cdot b\_n\}$, denoted by $T\_n$.

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