
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, one of its foci and the top and bottom vertices form a right-angled triangle. A circle with the diameter equal to the length of the major axis of the ellipse $C$ is tangent to the line $x+y-2=0$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ Suppose a line passing through the right focus of the ellipse and not parallel to the $x$-axis intersects the ellipse $C$ at points $A$ and $B$. Investigate whether there exists a fixed point $E$ on the $x$-axis such that $\overrightarrow{EA} \cdot \overrightarrow{EB}$ is a constant value? If it exists, find the constant value and the coordinates of point $E$; if not, explain why.

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