
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 the sum of its first $n$ terms denoted by $S\_n$. The first term is $a\_1 = \frac{1}{3}$, and the common ratio is $q > 0$. The sequence formed by $S\_1 + a\_1$, $S\_3 + \frac{7}{2}a\_3$, and $S\_2 + a\_2$ is an arithmetic sequence.

1. Find $a\_n$.
2. Let $b\_n = \frac{1}{\log_{\frac{1}{3}} a\_n}$ and $c\_n = b\_n(b_{n + 1} - b_{n + 2})$. Find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted by $T\_n$.

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