
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ whose left focus $F_1$ coincides with the focus of the parabola $y^2 = -4\sqrt{3}x$, and a line $l$ passing through point $F_1$ intersects the ellipse at points $A$ and $B$. When line $l$ passes through one of the short axis endpoints of ellipse $C$, it is tangent to a circle centered at the origin $O$ with the eccentricity $e$ of the ellipse as its radius.
1. Find the equation of ellipse $C$;
2. Determine whether there exists a fixed point $M$ on the x-axis such that $\overrightarrow{AM} \cdot \overrightarrow{BM}$ is a constant value. If it exists, find the fixed point $M$ and the constant value; if not, explain why.

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