
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
In cube A-C<sub>1</sub> with an edge length of 1, M is on edge AB with AM=1/3, P is a moving point on face ABCD, and the difference between the square of the distance from P to line A<sub>1</sub>D<sub>1</sub> and the distance from P to point M is 1. Determine the equation of the curve on which point P lies.

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