
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
Given two points \( M(-1, 0) \) and \( N(1, 0) \), and a point \( P \) such that \(\overrightarrow{MP} \cdot \overrightarrow{MN}\), \(\overrightarrow{PM} \cdot \overrightarrow{PN}\), and \(\overrightarrow{NM} \cdot \overrightarrow{NP}\) form an arithmetic sequence with a common difference less than zero, find the locus of point \(P\).

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