
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 the sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, and $a\_n = 1 - 2S\_n$ for all $n \in \mathbb{N}^*$.

(I) Find the general formula for the $n$-th term of the sequence.

(II) If $b\_n = \log_{\frac{1}{3}} a_{2n-1}$, $c\_n = \frac{4n^2}{b\_n b_{n+1}}$, and $T\_n$ denotes the sum of the first $n$ terms of the sequence $\{c\_n\}$, find the largest integer $k$ not exceeding $T_{2016}$.

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