
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  $p\geq 5$  be a prime number, and set  $M=\{1,2,\cdots,p-1\}.$  Define   $$ T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}. $$   If  $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$  with  $0\leq k\leq p-1,$  where  $\left[\alpha\right]$  denotes the largest integer that does not exceed  $\alpha,$  determine the value of  $k.$ 

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