
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
**p1A** Positive reals  $x$ ,  $y$ , and  $z$  are such that  $x/y +y/x = 7$  and  $y/z +z/y = 7$ . There are two possible values for  $z/x + x/z;$  find the greater value.**p1B** Real values  $x$  and  $y$  are such that  $x+y = 2$  and  $x^3+y^3 = 3$ . Find  $x^2+y^2$ .**p2** Set  $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$ . Let  $\sum S$  denote the sum of the members of  $S$ ; then  $\sum A = 149$ . Find the number of (not necessarily proper) subsets  $B$  of  $A$  for which  $\sum B \ge 75$ .**p3**  $99$  dots are evenly spaced around a circle. Call two of these dots ”close” if they have  $0$ ,  $1$ , or  $2$  dots between them on the circle. We wish to color all  $99$  dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than  $4$  different colors?**p4** Given a  $9 \times 9$  grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid.


PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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