
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 the origin \( O \), with foci on the \( y \)-axis. The distance from a focus to the corresponding directrix and the eccentricity are both \( \frac{\sqrt{2}}{2} \). The line \( l \) intersects the \( y \)-axis at point \( P(0, m) \) and intersects ellipse \( C \) at two distinct points \( A \) and \( B \), such that \( \overrightarrow{A P}=\lambda \overrightarrow{P B} \). 

1. Find the equation of the ellipse.
2. If \( \overrightarrow{O A}+\lambda \overrightarrow{O B}=4 \overrightarrow{O P} \), find the range of values for \( m \).

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