
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 the sequence {a<sub>n</sub>} is an arithmetic sequence, a<sub>n</sub>≠0, and it satisfies 32a<sub>3</sub>+32a<sub>11</sub>=a<sub>7</sub><sup>2</sup>, the sequence {b<sub>n</sub>} satisfies b<sub>n+1</sub>-2b<sub>n</sub>=0, b<sub>7</sub>=a<sub>7</sub>.
(Ⅰ) Find the general term formula for the sequence {b<sub>n</sub>};
(Ⅱ) If c<sub>n</sub>=nb<sub>n</sub>, find the sum of the first n terms S<sub>n</sub> for the sequence {c<sub>n</sub>}.

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