
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
Suppose you have a sphere tangent to the  $xy$ -plane with its center having positive  $z$ -coordinate.  If it is projected from a point  $P=(0,b,a)$  to the  $xy$ -plane, it gives the conic section  $y=x^2$ .  If we write  $a=\tfrac pq$  where  $p,q$  are integers, find  $p+q$ .

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