
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
p1. Given the set  $H = \{(x, y)|(x -y)^2 + x^2 - 15x + 50 = 0$  where x and y are natural numbers  $\}$ .
Find the number of subsets of  $H$ .


p2. A magician claims to be an expert at guessing minds with following show. One of the viewers was initially asked to hidden write a five-digit number, then subtract it with the sum of the digits that make up the number, then name four of the five digits that make up the resulting number (in order of any). Then the magician can guess the numbers hidden. For example, if the audience mentions four numbers result:  $0, 1, 2, 3$ , then the magician will know that the hidden number is  $3$ .
a. Give an example of your own from the above process.
b. Explain mathematically the general form of the process.


p3. In a fruit basket there are  $20$  apples,  $18$  oranges,  $16$  mangoes,  $10$  pineapples and  $6$  papayas. If someone wants to take  $10$  pieces from the basket. After that, how many possible compositions of fruit are drawn?


p4. Inside the Equator Park, a pyramid-shaped building will be made with base of an equilateral triangle made of translucent material with a side length of the base  $8\sqrt3$  m long and  $8$  m high. A globe will be placed in a pyramid the. Ignoring the thickness of the pyramidal material, determine the greatest possible length of the radius of the globe that can be made.


p5. What is the remainder of  $2012^{2012} + 2014^{2012}$  divided by  $2013^2$ ?

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