
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
Find the general formula for the sequence $\{n\}$,  
Given that the sum of the first $n$ terms of sequence $\{a_n\}$ is $S_n$, and vectors $\overrightarrow{a} = (S_n, 1)$, $\overrightarrow{b} = (2n - 1, \frac{1}{2})$ satisfy the condition $\overrightarrow{a} \parallel \overrightarrow{b}$,  
Find the general formula for the sequence $\{n\}$,  
Let $n = \frac{b_{n}}{a_{n}}$, find the sum of the first $n$ terms of sequence $\{c_n\}$, denoted as $T_n$.

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