
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
Let \(\mathrm{p} > 3\) be a prime number, and define the set \(M = \{1, 2, \ldots, p-1\}\). Consider the set 
\[ T = \left\{\left(n, x_{n}\right) : p \mid (n x_{n} - 1) \text{, and } n, x_{n} \in M\right\}. \]
Find the smallest non-negative residue modulo \(\mathrm{p}\) of 
\[ \sum_{\left(\mathrm{n}, x_{n}\right) \in \mathrm{T}} \mathrm{n}\left\lfloor \frac{\mathrm{nx}}{\mathrm{p}} \right\rfloor. \]

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