
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 that there exists $x_{0} \in \mathbb{R}$ such that the inequality $|x-1|-|x-2| \geqslant t$ holds.  
$(1)$ Find the set $T$ of real numbers $t$ that satisfy the condition;  
$(2)$ If $m > 1$ and $n > 1$, for $\forall t \in T$, the inequality $\log_{3}m \cdot \log_{3}n \geqslant t$ always holds, find the minimum value of $mn$.

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