
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 that the ellipse $C\_1$: $\frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1 (a > b > 0)$ and the parabola $C\_2$: $x^{2}=2py (p > 0)$ share a common focus, and the directrix $l$ of the parabola $C\_2$ intersects with the ellipse $C\_1$ at the point $(\sqrt{2},-2)$.

1. Find the equations of the ellipse $C\_1$ and the parabola $C\_2$.
2. If point $P$ is a moving point on line $l$, draw two tangents to the parabola through point $P$, with the tangent points being $A$ and $B$. The line $AB$ intersects the ellipse $C\_1$ at points $E$ and $F$. Find the range of values for $\overrightarrow{OE} \cdot \overrightarrow{OF}$.

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