
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 $\{a\_n\}$ with a non-zero common difference, $a\_1$, $a\_2$, and $a\_5$ form a geometric sequence. The sum of the first 10 terms of this sequence is 100. The sequence $\{b\_n\}$ has a sum of its first n terms denoted as $S\_n$, satisfying $S\_n = 2b\_n - 1$ for all positive integers $n$.

(I) Find the general term formulas for the sequences $\{a\_n\}$ and $\{b\_n\}$.
(II) Let $c\_n = \frac{1+a\_n}{4b\_n}$. Find the range of possible values for $T\_n$, the sum of the first n terms of the sequence $\{c\_n\}$.

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