
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 a point M on the ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ ($a > b > 0$), a circle with center M touches the x-axis at the right focus F of the ellipse.
(1) If the circle M is tangent to the y-axis, find the eccentricity of the ellipse.
(2) If the circle M intersects the y-axis at points A and B, and $\triangle ABM$ is an equilateral triangle with side length 2, find the equation of the ellipse.

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