
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
 $(BEL 1)$  A parabola  $P_1$  with equation  $x^2 - 2py = 0$  and parabola  $P_2$  with equation  $x^2 + 2py = 0, p > 0$ , are given. A line  $t$  is tangent to  $P_2.$  Find the locus of pole  $M$  of the line  $t$  with respect to  $P_1.$ 

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