
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 positive geometric sequence \\(\{a_n\}\) satisfying that \\(a_1\), \\(2a_2\), \\(a_3+6\) form an arithmetic sequence, and \\(a_4^2=9a_1a_5\),  
\\((I)\\) find the general formula for the sequence \\(\{a_n\}\);  
\\((II)\\) let \\(b_n=(\log_{\sqrt{3}}a_n+1)\cdot a_n\), find the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\), denoted as \\(T_n\).

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