
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 sequence \\(\{a_n\}(n=1,2,3,\ldots)\\) have a sum of the first \\(n\\) terms \\(S_n\\) that satisfies \\(S_n=2a_n-a_1\\), and \\(a_1\\), \\(a_2+1\\), \\(a_3\\) form an arithmetic sequence.  
(Ⅰ) Find the general formula for the sequence \\(\{a_n\}\\).  
(Ⅱ) 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\\) such that \\(\left|T_n-1\right| < \frac{1}{1000}\\) holds.

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