
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 $\{a\_n\}$ whose first $n$ terms' sum $S\_n$ satisfies the equation $S\_n = p(S\_n - a\_n) + \frac{1}{2}$ ($p$ is a constant greater than 0), and $a\_1$ is the arithmetic mean of $6a\_3$ and $a\_2$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. If $a\_n \cdot b\_n = 2n + 1$, find the sum of the first $n$ terms, denoted as $T\_n$, for the sequence $\{b\_n\}$.

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