
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 the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ has its left focus $F$ as the center of the circle $x^2 + y^2 + 2x = 0$, and the minimum distance from a point on the ellipse to point $F$ is $\sqrt{2} - 1$.

$(1)$ Find the equation of the ellipse;

$(2)$ Given a moving line $l$ passing through point $F$ intersects the ellipse at two distinct points $A$ and $B$, and point $M\left(-\dfrac{5}{4}, 0\right)$, find the value of $\overrightarrow{MA} \cdot \overrightarrow{MB}$.

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