
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 $x \in \mathbb{R}$, find the set of real numbers $t$ such that the inequality $|x-1|-|x-2| \geqslant t$ always holds.

(1) Determine the set $T$ consisting of such real numbers $t$.

(2) If $m > 1$, $n > 1$, and for all $t \in T$, the inequality $\log_{3}(m) \cdot \log_{3}(n) \geqslant t$ always holds, find the minimum value of $m+n$.

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