
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
**p16.** When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is  $s(t) = t^3 - 3t^2$ . His climbing velocity as a function of the strength in his arms is  $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$ . At how many (possibly negative) points in time is Eric stationary?**p17**. Consider a triangle  $\vartriangle ABC$  with angles  $\angle ACB = 60^o$ ,  $\angle ABC = 45^o$ . The circumcircle around  $\vartriangle ABH$ , where  $H$  is the orthocenter of  $\vartriangle ABC$ , intersects  $BC$  for a second time in point  $P$ , and the center of that circumcircle is  $O_c$ . The line  $PH$  intersects  $AC$  in point  $Q$ , and  $N$  is center of the circumcircle around  $\vartriangle AQP$ . Find  $\angle NO_cP$ .**p18.** If  $x, y$  are positive real numbers and  $xy^3 = \frac{16}{9}$  , what is the minimum possible value of  $3x + y$ ?


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

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