
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 that $\{a\_n\}$ is an arithmetic sequence with a common difference of $d$, and the sum of its first $n$ terms is $S\_n$, where $S\_4 = 2S\_2 + 8$.
(I) Find the value of the common difference $d$;
(II) If $a\_1 = 1$, let $T\_n$ be the sum of the first $n$ terms of the sequence $\{\frac{1}{a\_n a\_{n+1}}\}$. Find the largest positive integer $m$ such that the inequality $T\_n \geq \frac{1}{18}(m^2 - 5m)$ holds true for all $n \in \mathbb{N}^*$.

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