
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 an ellipse $D: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ with an eccentricity of $e= \frac{\sqrt{2}}{2}$, and a point $(\sqrt{2},-1)$ on the ellipse $D$.

1. Find the standard equation of the ellipse $D$.
2. A line $l$ passes through a point $E(0,t)$ on the y-axis with a slope of $k$, intersecting the ellipse at points $A$ and $B$. The slopes of lines $OA$ and $OB$ (where $O$ is the origin) are denoted as $k_{OA}$ and $k_{OB}$, respectively. If there exists a $\lambda \in [2,4]$ such that $k_{OA} + k_{OB} = \lambda k$ for any real number $k$, find the range of values for the real number $t$.

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