
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
 $n$  students take a test with  $m$  questions, where  $m,n\ge 2$  are integers. The score given to every question is as such: for a certain question, if  $x$  students fails to answer it correctly, then those who answer it correctly scores  $x$  points, while those who answer it wrongly scores  $0$ . The score of a student is the sum of his scores for the  $m$  questions. Arrange the scores in descending order  $p_1\ge p_2\ge \ldots \ge p_n$ . Find the maximum value of  $p_1+p_n$ . 

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