
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
The center of ellipse $C$ is at the origin $O$, its foci are on the $y$-axis, and both the distance from the foci to the corresponding directrix and the eccentricity are $\frac{\sqrt{2}}{2}$. The line $l$ intersects the $y$-axis at point $P(0, m)$ and intersects the ellipse $C$ at two distinct points $A$ and $B$, with $\overrightarrow{AP} = \lambda \overrightarrow{PB}$.
(1) Find the equation of the ellipse;
(2) If $\overrightarrow{OA} + \lambda \overrightarrow{OB} = 4\overrightarrow{OP}$, find the range of values for $m$.

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