
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**. Your Halloween took a bad turn, and you are trapped on a small rock above a sea of lava. You are on rock  $1$ , and rocks  $2$  through  $12$  are arranged in a straight line in front of you. You want to get to rock  $12$ . You must jump from rock to rock, and you can either (1) jump from rock  $n$  to  $n + 1$  or (2) jump from rock  $n$  to  $n + 2$ . Unfortunately, you are weak from eating too much candy, and you cannot do (2) twice in a row. How many different sequences of jumps will take you to your destination?**p2.** Find the number of ordered triples  $(p; q; r)$  such that  $p, q, r$  are prime,  $pq + pr$  is a perfect square and  $p + q + r \le 100$ .**p3.** Let  $x, y, z$  be nonzero complex numbers such that  $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} \ne 0$  and  $$ x^2(y + z) + y^2(z + x) + z^2(x + y) = 4(xy + yz + zx) = -3xyz. $$   Find  $\frac{x^3 + y^3 + z^3}{x^2 + y^2 + z^2}$  .


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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