
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 $C: \frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a>b>0$) with a focal distance of 2, and one endpoint of the minor axis of the ellipse $C$ forms an equilateral triangle with the left and right foci $F_1$ and $F_2$.
(1) Find the standard equation of the ellipse $C$;
(2) Let $M$ be any point on the ellipse $C$, find the maximum and minimum values of $\overrightarrow{MF_{1}} \cdot \overrightarrow{MF_{2}}$;
(3) Determine whether there exists a point $B$ on the x-axis such that for any point $P$ on the ellipse, the ratio of the distance from $P$ to $B$ and the distance from $P$ to the line $x = 4$ is constant. If such a point exists, find the coordinates of point $B$; if not, explain why.

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