
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 the Cartesian coordinate plane, points $A\_1(-\sqrt{2},0)$, $A\_2(\sqrt{2},0)$, $P(x,y)$, $M(x,1)$, and $N(x,-2)$ are given. If there exists a positive real number $\lambda$ such that $\lambda^{2}\overrightarrow{{OM}} \cdot \overrightarrow{{ON}} = \overrightarrow{A\_1P} \cdot \overrightarrow{A\_2P}$ (where $O$ is the coordinate origin), find the trajectory equation of point $P$ and discuss the type of the trajectory.

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