
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 that ${{a_{n}}}$ is a geometric sequence with all positive terms, (${a_{1} = 1, a_{5} = 256}$), and ${S_{n}}$ is the sum of the first $n$ terms of the arithmetic sequence ${{b_{n}}}$ (${b_{1} = 2, 5S_{5} = 2S_{8}}$).

(I) Find the general term formulas for both ${{a_{n}}}$ and ${{b_{n}}}$.

(II) Let ${T_{n} = a_{1}b_{1} + a_{2}b_{2} + \cdots + a_{n}b_{n}}$, find ${T_{n}}$.

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