
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 an arithmetic sequence {a<sub>n</sub>} that is an increasing sequence with a common ratio of q, the sum of the first n terms is S<sub>n</sub>. The sequence satisfies the conditions a<sub>2</sub> + a<sub>3</sub> + a<sub>4</sub> = 28 and a<sub>3</sub> + 2 is the arithmetic mean of a<sub>2</sub> and a<sub>4</sub>.
(I) Find the general term formula for the sequence {a<sub>n</sub>}.
(II) If b<sub>n</sub> = a<sub>n</sub> · log<sub>2</sub> (1/a<sub>n</sub>) and T<sub>n</sub> = b<sub>1</sub> + b<sub>2</sub> + ... + b<sub>n</sub>, find the positive integer value of n that makes T<sub>n</sub> + n · 2^(n+1) = 30 true.

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