
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 $E$ with the equation $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a > b > 0$), the distance from its right focus to the line $x - y + 2\sqrt{2}=0$ is $3$, and the ellipse passes through the point $(-1, - \frac{\sqrt{6}}{2})$.
(1) Determine the equation of ellipse $E$.
(2) Let the left vertex of ellipse $E$ be $A$. A line $l$ with the equation $x-my-t=0$ intersects the ellipse $E$ at two distinct points $M$ and $N$ (both different from $A$). A circle with diameter $MN$ passes through point $A$. Determine whether the line $l$ passes through a fixed point; if it does, find the coordinates of that fixed point.

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