
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 a sequence $\{ a\_n \}$ with its first $n$ terms sum $S\_n$, the sequence $\{ \frac{S\_n}{n} \}$ is an arithmetic sequence with first term $1$ and common difference $2$.

(1) Find the general formula for the sequence $\{ a\_n \}$;
(2) Let sequence $\{ b\_n \}$ satisfy $b\_n = \frac{1}{a\_n \cdot a\_{n+1}}$, find the sum of the first $n$ terms of sequence $\{ b\_n \}$, denoted as $T\_n$;
(3) If the positive geometric sequence $\{ c\_n \}$ satisfies $c\_1 = a\_1$ and $c\_2$ is the geometric mean of $a\_1$ and $a\_3$, find the sum of the first $n$ terms of sequence $\{ c\_n \}$, denoted as $C\_n$.

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