
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
The function y $ = x^2$  is graphed in the  $xy$ -plane. A line from every point on the parabola is drawn to the point  $(0,-10, a)$  in three-dimensional space. The locus of points where the lines intersect the  $xz$ -plane forms a closed path with area  $\pi$ . Given that  $a = \frac{p\sqrt{q}}{r}$  for positive integers  $p$ ,  $q$ , and  $r$  where  $p$  and  $r$  are relatively prime and  $q$  is not divisible by the square of any prime, find  $p + q + r$ .

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