
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
If  $p$  is a prime positive integer and   $x,y$  are positive integers, 
find , in terms of  $p$ , all pairs  $(x,y)$  that are solutions of the equation:  $p(x-2)=x(y-1)$ . (1)
If it is also given that  $x+y=21$ , find all triplets   $(x,y,p)$  that are solutions to equation (1).

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