
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 massive school which has  $m$  students, and each student took at least one subject. Let  $p$  be an odd prime. Given that:

(i) each student took at most  $p+1$  subjects. 
(ii) each subject is taken by at most  $p$  students. 
(iii) any pair of students has at least  $1$  subject in common. 

Find the maximum possible value of  $m$ .

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