
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 geometric sequence $\{a_n\}$ with common ratio $q$, the product of the first $n$ terms is denoted as $T_n$, and the following conditions are met: $a_1 > 1$, $a_{99}a_{100} - 1 > 0$, and $\frac{a_{99}-1}{a_{100}-1} < 0$. Consider the statements: $①\ 0 < q < 1$; $②\ a_{99}a_{101} - 1 < 0$; $③\ T_{100}$ is the maximum value of $T_n$; $④$ the largest natural number $n$ such that $T_n > 1$ is equal to $198.$ Determine which of the statements are correct.

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