
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 system, given points $A_1(-\sqrt{2},0)$, $A_2(\sqrt{2},0)$, $P(x,y)$, $M(x,1)$, $N(x,-2)$, find the real number $\lambda$ such that $\lambda^2 \overrightarrow{OM} \cdot \overrightarrow{ON} = \overrightarrow{A_1P} \cdot \overrightarrow{A_2P}$ (where O is the origin).
(1) Determine the equation of point P's locus and discuss the type of trajectory it represents.
(2) When $\lambda = \dfrac{\sqrt{2}}{2}$, if a line l passing through point B(0,2) intersects with the locus of point P at two distinct points E and F (E lies between B and F), find the range of the ratio of the areas of triangles OBE and OBF.

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