
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
Consider an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ ($a>b>0$), where its right vertex is point A and its lower vertex is point B. A circle passing through points A, O (the coordinate origin), and B has its center at coordinates $(\frac{\sqrt{3}}{2}, -\frac{1}{2})$.

1. Find the equation of the ellipse.
2. Given that point M lies on the positive semi-axis of x, and the perpendicular line drawn from point B through M intersects the ellipse at another point N such that $\angle{BMN} = 60^{\circ}$, find the coordinates of point M.

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