
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
The common ratio of the geometric sequence $\{a_n\}$ is $q$, and the product of the first $n$ terms is $T_n$. It is also given that $a_1 > 1$, $a_{2009} \cdot a_{2010} - 1 > 0$, and $(a_{2009} - 1)(a_{2010} - 1) < 0$. The following conclusions are drawn:
① $0 < q < 1$;
② $a_{2009} \cdot a_{2011} < 1$;
③ $T_{2010}$ is the largest among all $T_n$;
④ The largest natural number that makes $T_n > 1$ true is 4018.
Among these conclusions, the correct ones are.

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