
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]


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In 1765, the famous Swiss mathematician Euler proposed a theorem: the circumcenter, centroid, and orthocenter of a triangle are collinear, and this line is called the "Euler line" of the triangle. In the Cartesian coordinate system, $\triangle ABC$ satisfies $AC=BC$, with vertices $A(-1,0)$, $B(1,2)$, and its "Euler line" tangent to the circle $M: (x+5)^2 + y^2 = r^2 (r > 0)$.  
$(1)$ Find the equation of the "Euler line" of $\triangle ABC$.  
$(2)$ If the circle $M$ has common points with the circle $x^2 + (y-a)^2 = 2$, find the range of $a$.  
$(3)$ If the point $(x,y)$ lies on the "Euler line" of $\triangle ABC$, find the minimum value of $\sqrt{x^2 + y^2 - 2x - 2y + 2} + \sqrt{(x-2)^2 + y^2}$.

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