
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 for \\(x\\).
\\((I)\\) Find the set of real numbers \\(T\\) that satisfies the condition;
\\((II)\\) If \\(m > 1\\) and \\(n > 1\\), and for every \\(t \in T\\), the inequality \\(\log_{3}m \cdot \log_{3}n \geqslant t\\) always holds, try to find the minimum value of \\(m+n\\).

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