
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
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n=2a_n-a_1$, and $a_1$, $a_2+1$, $a_3$ form an arithmetic sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_n} \right\}$ be $T_n$, find the minimum value of $n$ for which $|T_n-1| < \frac{1}{1000}$ holds.

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