
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 sum of the first $n$ terms is $S\_n = 2a_n - a_1$, and $a\_1$, $a\_2 + 1$, $a\_3$ form an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $T\_n$ denote the sum of the first $n$ terms of the sequence $\{ \frac{1}{a\_n} \}$. Find the smallest value of $n$ that makes $|T\_n - 1| < \frac{1}{2016}$ hold true.

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