
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  $p$ and  $q=p+2$  be twin primes. consider the diophantine equation  $(+)$  given by 
                  $n!+pq^2=(mp)^2$   $m\geq1$ ,  $n\geq1$ i. if   $m=p$ ,find the value of  $p$ .
ii. how many solution quadruple  $(p,q,m,n)$  does  $(+)$  have ?

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