
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 its left focus $F_1$ coinciding with the focus of the parabola $y^2 = -4\sqrt {3}x$, a line $l$ passing through point $F_1$ intersects the ellipse at points A and B. When line $l$ passes through one endpoint of the minor axis of the ellipse C, it is tangent to a circle centered at the origin O with a radius equal to the eccentricity $e$ of the ellipse.
(1) Find the equation of ellipse C;
(2) Are there any fixed points M on the x-axis such that the dot product $\overrightarrow {AM} \cdot \overrightarrow {BM}$ is a constant value? If there is such a point M, find the point and the constant value; if it doesn't exist, explain why.

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