
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
Selected Exercise $(4-5)$: Inequality Lecture

Given that $\exists x \in R$ makes the inequality $|x-1|-|x-2|\geqslant t$ hold.

(1) Find the set $T$ of real numbers $t$ that satisfy the condition;

(2) If $m > 1$, $n > 1$, and the inequality ${\log_3}m \cdot {\log_3}n \geqslant t$ always holds for $\forall t \in T$, find the minimum value of ${m^2} + {n^2}$.

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