
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 a common ratio $q \neq 1$, the first term $a\_1 = \frac{1}{3}$, and $a\_1, 2a\_2, 3a\_3$ forms an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$;
2. Find the sum $T\_n$ of the first $n$ terms of the sequence $\{\frac{n}{a\_n}\}$;
3. If $c\_n = \log_{\frac{1}{3}} a_{2n-1}$, and $P\_n$ is the sum of the first $n$ terms of the sequence $\{\frac{4n^2}{c\_n c\_{n+1}}\}$, find the largest integer $k$ not exceeding $P\_{2016}$.

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