
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
**p17.** Let the roots of the polynomial  $f(x) = 3x^3 + 2x^2 + x + 8 = 0$  be  $p, q$ , and  $r$ . What is the sum  $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$  ?**p18.** Two students are playing a game. They take a deck of five cards numbered  $1$  through  $5$ , shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play?**p19.** Compute the sum of all primes  $p$  such that  $2^p + p^2$  is also prime.**p20.** In how many ways can one color the  $8$  vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color?


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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