
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 increasing arithmetic sequence $\{a_n\}$ which satisfies: $a_1, a_2, a_4$ form a geometric sequence, and $a_1 \neq 1$.
(I) Find the general term formula for the sequence $\{a_n\}$;
(II) If $b_n = \log_2 \left(1 + \frac{1}{a_n}\right)$, and $T_n = b_1 + b_2 + \ldots + b_n$, find the sum of the first n terms $S_n$ for the sequence $\left\{ \frac{1}{2^{T_n} \cdot 2^{T_{n+1}}} \right\}$.

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