
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
In a test, there are $m$ questions and $n$ students taking part, where $m$ and $n$ are given integers with $m, n \geqslant 2$. The scoring rule for each question is: if exactly $x$ students do not answer the question correctly, then each student who answers the question correctly receives $x$ points, while those who do not answer it correctly receive zero points. Each student's total score is the sum of their scores from all $m$ questions. If the total scores of all students are sorted in descending order as $p_{1} \geqslant p_{2} \geqslant \cdots \geqslant p_{n}$, find the maximum possible value of $p_{1}+p_{n}$.

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