
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 of positive terms $\{a\_n\}$ has a sum of $S\_n$ for the first $n$ terms, if both $\{a\_n\}$ and $\{\sqrt{S\_n}\}$ are arithmetic sequences with equal common differences:

1. Find the general term formula for the sequence $\{a\_n\}$;
2. Let $b\_n = \frac{1}{4a\_n}$ and $c\_n = b\_n \cdot b\_{n+1}$, find the sum $T\_n$ of the first $n$ terms for the sequence $\{c\_n\}$.

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