
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 an ellipse $C$ is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$, passes through the point $(\sqrt{2},1)$, and has a focal length of $2\sqrt{2}$.

1. Find the equation of ellipse $C$.
2. Suppose a line $L: y = k(x+1)$ where $k > -2$, intersects ellipse $C$ at two distinct points $A$ and $B$. The midpoint of line segment $AB$, denoted as $M$, has a distance of $\frac{3\sqrt{5}}{5}$ to the line $2x + y + t = 0$. Determine the range of the real number $t$ where $t > 2$.

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