
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
Construct equilateral triangles $BCA'$, $CAB'$, and $ABC'$ externally on the sides $BC$, $CA$, and $AB$ of triangle $ABC$, respectively. The lines $AA'$, $BB'$, and $CC'$ intersect at a point, which is called the circumcenter of triangle $ABC$. When each interior angle of triangle $ABC$ is less than $120^{\circ}$, the circumcenter $P$ of triangle $ABC$ satisfies the following properties:  
$(1)$ $\angle APB=\angle APC=\angle BPC=120^{\circ}$;  
$(2)$ The circumcenter is the point with the minimum sum of distances to the three vertices of the triangle (also known as the Fermat point).  
Given the properties above, the minimum value of $\sqrt{{x}^{2}+(y-1)^{2}}+\sqrt{{x}^{2}+(y+1)^{2}}+\sqrt{(x-2)^{2}+{y}^{2}}$ is ____.

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