
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 sequence $\left\{ a_n \right\}$ is a geometric sequence with a common ratio of $q$, its sum of the first $n$ terms is $S_n$, and the product of the first $n$ terms is $T_n$. Given that $0 < a_1 < 1, a_{2012}a_{2013} = 1$, the correct conclusion(s) is(are) ______.

$(1) q > 1$ $(2) T_{2013} > 1$  $(3) S_{2012}a_{2013} < S_{2013}a_{2012}$   $(4)$ The smallest natural number $n$ for which $T_n > 1$ is $4025$    $(5) \min \left( T_n \right) = T_{2012}$

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