
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
The right focus of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ is $F_1(2,0)$, and a tangent line to the circle $x^2 + y^2 = b^2$ passing through $F_1$ intersects the $y$-axis at point $Q$. The tangent point $N$ is the midpoint of the line segment $F_1Q$.

(Ⅰ) Find the equation of the ellipse;

(Ⅱ) If the curve $y = x^2 + m$ intersects the ellipse at four points, and these four points are all on the same circle, find the coordinates of the center of this circle.

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